text:
\section{Introduction} Our motivation is the celebrated result by Wente \cite{Wente80}, which shows that a constant mean curvature hypersurface with constant contact angle along an hyperplane is a spherical cap. The more general statement contained in Wente's paper is actually concerned with hypersurfaces whose mean curvature depends only on the distance from their bounding hyperplane. More precisely, in \cite{Wente80}, it was proven that:

\medskip

\noindent {\it Let $E\subset\mathbb R^n$, $n\ge 2$, be a bounded open connected subset of $H=\{x_n>0\}$ such that $M=\overline{H\cap\partial E}$ is a $C^2$-hypersurface with boundary ${\rm bd}(M)=M\cap\partial H$. If, for a suitable constant $\sigma\in(-1,1)$ and function $g:(0,\infty)\to\mathbb R$,
\begin{equation*}\begin{split}
&\mathbf{H}_E(q)=g(q_n)\qquad\forall q\in M\cap H\,,
\\
&\nu_E(q)\cdot e_n=\sigma\qquad\forall q\in M\cap\partial H\,,
\end{split}\end{equation*}
then $\{M\cap\{x_n=t\}:t>0\}$ is a family of $(n-2)$-dimensional spheres centered on the same vertical axis. Here $\overline{A}$ and $\partial A$ denote the topological closure and the topological boundary of any $A\subset \mathbb R^n$, ${\rm bd}\,(M)$ is the manifold boundary, and $\mathbf{H}_E$ is the mean curvature of $M$ computed with respect to the outer unit normal $\nu_E$ to $E$.}

\medskip

We want to prove a generalization of this theorem with the classical mean curvature replaced by the fractional mean curvature, which was introduced in \cite{caffaroquesavin} as the first variation of fractional perimeter (see the discussion below Theorem \ref{thm axially symmetric} for additional context). Let us recall that, if $E$ is an open subset of $\mathbb R^n$, then the fractional mean curvature of order $s\in(0,1)$ is defined for each $q\in\partial E$ as
\begin{equation}
\label{fractional mean curvature}
\mathbf{H}^{s}_{E} (q):={\rm p.v.}\,\int_{\mathbb R^n}\frac{\chi_{E^c}(x)-\chi_{E}(x)}{|x-q|^{n+s}}dx={\rm p.v.}\,\int_{\mathbb R^n}\frac{\tilde{\chi}_{E}(x)}{|x-q|^{n+s}}dx\,,
\end{equation}
where $\chi_E$ is the characteristic function of $E$ and
\[
\tilde{\chi}_E:=\chi_{E^c}-\chi_E\,.
\]
 The integral in \eqref{fractional mean curvature} converges at $q\in\partial E$ as soon as $E$ is of class $C^{1,\a}$ around $q$ for some $\a\in(s,1)$. Indeed, in this case, given $r>\varepsilon>0$, the factor $\tilde{\chi}_{E}$ allows one to localize the integral of $|x-q|^{-n-s}$ over $B_r(q)\setminus B_\varepsilon(q)$ to a smaller region of the form $P\setminus B_\varepsilon(q)$, where $P\subset B_r(q)$ is a set enclosed between two tangent $C^{1,\a}$-paraboloids. For a set $E$ as in Wente's theorem, the size of the region where this kind of cancellation is possible becomes increasingly small as $q$ approaches $\partial H$, and as a consequence $\mathbf{H}^s_{E}(q)$ will blow-up as $q_n\to 0^+$, at a rate defined by the contact angle. More precisely, as we show in Proposition \ref{H blow}, if $E$ satisfies the fractional variant of the assumptions of Wente's theorem, then
\begin{equation}\label{blow up}
\mathbf{H}_E^s(q)=\frac{c(n,s,\sigma)}{q_n^s}\Big(1+{\rm O}(1)\Big)\qquad\mbox{as $q_n\to 0^+$ with $q\in \partial E\cap H$}\,.
\end{equation}
This singular behavior is an unavoidable and challenging feature of extending Wente's result to the fractional setting; overcoming it will be the most interesting point in the proof of the main result.

To state our result we define the $s$-deficit of $E$
\[
\delta_s(E)=\mathrm{diam}(E)^{s+1}\sup\limits_{p_n=q_n \atop p,q\in M\cap H} \frac{|\mathbf{H}_E^s(p)-\mathbf{H}_E^s(q)|}{|p-q|^{}}\,,
\]
to measure how far away $\mathbf{H}_E^s$ is from being constant on horizontal slices in $H$. Note that in this definition the $\mathrm{diam}(E)^{s+1}$ term is used to enforce that $\delta_s(E)$ is scale invariant. The following theorem explains how, for small $\delta_s,$ we have that $E$ is almost axially symmetric with respect to $\delta_s$:

\begin{theorem}\label{thm axially symmetric}
Let $s\in(0,1)$, $\a\in(s,1)$, and let $E\subset\mathbb R^n$, $n\ge 2$, be a bounded open connected subset of $H=\{x_n>0\}$ such that $M=\overline{\partial E\cap H}$ is a $C^{2,\a}$-hypersurface with boundary ${\rm bd}(M)=M\cap\partial H$. Let  $\sigma\in(-1,1)$ such that
\begin{equation}\label{sigma}
\begin{split}
&\nu_E(q)\cdot\nu_H(q)=\sigma\qquad\forall q\in M\cap\partial H\,.
\end{split}
\end{equation}
\begin{enumerate}[label=(\alph*)]
\item For every direction $e\in S^{n-1}\cap\partial H$ there is a hyperplane $\pi_e$ orthogonal to $e$ such that, if we let $\rho$ denote reflection across $\pi_{e}$, then there exists a constant $C=C(n,s)$ dependent only on $n$ and $s$, such that
\begin{equation*}
|E\Delta\rho (E)|\le C\mathrm{diam}(E)^{n}\sqrt{\delta_s(E)}\,.
\end{equation*}
\smallskip
\item\label{improved thm} For $h>0$ such that $E_h=\{x\in E:x_n=h\}\ne \emptyset,$ set
\[
r_h=\inf\limits_{x\in {\rm bd}\, (E_h)}|x-h e_n|,\qquad R_h=\sup\limits_{x\in {\rm bd}\,( E_h)}|x-h e_n|\,,
\]
and
\[
D_{h}:=B_{r_h}(he_n)\cap \{x:x_n=h\}\,
\]
There exist $\delta_0=\delta_0(n,s)$ and $C_0=C_0(n)$ such that, if $\delta_s(E)\le \delta_0,$ then
\begin{equation}\label{R diff bound}
\frac{R_h-r_h}{\mathrm{diam}(E)}\le 2C_0\frac{\mathrm{diam}(E)^{n}}{|E|}\sqrt{\delta_s(E)} ,\qquad \forall h\in (0,\sup\{x_n:x\in E\})\,,
\end{equation}
up to horizontal translations.
Moreover, if
\[
\mathrm{diam}(E_h)> 6C_0\frac{\mathrm{diam}(E)^{n+1}}{|E|}\sqrt{\delta_s(E)}\,,
\]
then
\[
D_{h}\subset E_h\,.
\]
or, if not, then
\begin{equation}\label{Rh bound}
\frac{R_h}{\mathrm{diam}(E)}\le 7C_0\frac{\mathrm{diam}(E)^{n}}{|E|}\sqrt{\delta_s(E)}\,.
\end{equation}
\end{enumerate}

\end{theorem}
\begin{remark}{\rm The constants in the statement are computable and are included in the proof. }
\end{remark}

\begin{remark} {\rm The second part  of the theorem implies that for a small enough cross-sectional diameter, we do not expect that the cross section is centered around the $x_n$-axis, but rather that it is contained in a small ball around the axis. However, if the cross section has large enough diameter in terms of the deficit, then it is pinched between two balls of close radii, both centered around the $x_n$-axis.
\end{remark}

\begin{remark}
{\rm The term $\mathrm{diam}(E)^n/|E|$, in \eqref{R diff bound} and \eqref{Rh bound}, is scale invariant and might be unbounded for a sequence of $E_m$ satisfying the hypotheses of the theorem. Indeed we could consider a sequence of elongated ellipses of fixed volume, for which this ratio does explode. However, for classical perimeter, there are density estimates that, when combined with a perimeter restriction, imply that in the case of small mean curvature deficit the ratio $\mathrm{diam}(E)^n/|E|$ can be bounded above by a constant. It could be interesting to see if, under assumptions on the contact angle between $M$ and $\partial H$, there are density estimates along the lines of those in \cite{caffaroquesavin, millotsirewang} that would provide an upper bound in terms of $n$ and $P_s(E)$ for this term. }
\end{remark}

The statement of the theorem implies the following corollary:
\begin{corollary}\label{cor axially symmetric}
Let $s,\a,\sigma, E$ and $M$ be as in Theorem~\ref{thm axially symmetric}. If $\delta_s(E)=0,$ i.e. if,
\[
\mathbf{H}^s_E(q)=g(q_n)\,,
\]
for $q\in M\cap H$ and some function $g$ dependent only on the vertical direction, then $E$ is axially symmetric.
\end{corollary}

\begin{remark}
{\rm The behavior of $g(t)$ as $t\to 0^+$ in Corollary \ref{cor axially symmetric} is determined by $\sigma$ in \eqref{sigma} according to \eqref{blow up}.}
\end{remark}

We now give additional context to Theorem \ref{thm axially symmetric}. Our work joins the efforts of many authors in understanding geometric variational problems in the fractional setting. This line of research was initiated in \cite{caffaroquesavin} with the study of Plateau problem with respect to the fractional perimeter:
\[
P_{s}(E)=I_s(E,E^c)\,,
\]
where
\[
I_s(E,F)=\int_E\int_F\frac{dxdy}{|x-y|^{n+s}}
\]
is the the fractional interaction energy of a pair of disjoint sets $E,F\subset\mathbb R^n$. (For further studies of nonlocal minimal surfaces see for instance \cite{AmbdPhiMart, barriosfigallivald,figvald,savinvald, daviladelpinodipievald, davdelpwei}.)
The fractional mean curvature operator $\mathbf{H}^s_{E}$ defined in \eqref{fractional mean curvature} arises because, if $E$ is of class $C^2$ around a point $x\in\partial E$, then the first variation of $P_s(E)$ along the flow generated by a compactly supported and smooth vector-field $X$ satisfies
\[
\delta P_s(E)[X]=\int_{\partial E}\,(X\cdot\nu_E)\,\mathbf{H}^s_{E}\,d\mathcal H^{n-1}\,,
\]
(see \cite{figfuscmagmillmor, abatvald} for further results).
In this direction, the closest related result to Theorem~\ref{thm axially symmetric} is the recent extension of the classical rigidity theorem of Aleksandrov \cite{aleksandrov} (boundaries of compact sets with constant mean curvature are spheres) to fractional mean curvatures due to \cite{ciraolofigallimagginovaga,cabrefallweth1}. To be precise, in these papers it was shown that, if $E$ is a bounded open set with boundary of class $C^{2,\a}$ for some $\a\in(s,1)$ and such that $\mathbf{H}^s_{E}$ is constant along $\partial E$, then $\partial E$ is a sphere. This was proven by adapting the original moving plane argument by Aleksandrov to the fractional setting.

Following this classical moving plane method will also be our approach to Theorem~\ref{thm axially symmetric}. A moving hyperplane perpendicular to $\partial H$ is slid in a given direction until the reflected cap of $M$ across this hyperplane achieves a first contact point with $M$. In the well-known case of Aleksandrov's argument one has to discuss two kinds of tangency points, which become four different kinds of tangency points in Wente's work \cite{Wente80}, and also in our situation. Two of these four cases, where the tangency point is achieved away from $\partial H$, follow by repeating the arguments of \cite{ciraolofigallimagginovaga,cabrefallweth1}. However, the other two cases require new considerations because of the aforementioned degeneracy of the fractional mean curvature near the boundary hyperplane. Their proof is the main contribution of the paper.

As much as Wente's result is related to the study of classical capillarity theory, and in particular to the study of critical points for the Gauss free energy in the sessile droplet problem (see \cite{Finn} or \cite[Chapter 19]{maggiBOOK}), Theorem \ref{thm axially symmetric} can also be motivated by the study of a capillarity model using fractional perimeters to mimic surface tension. More precisely, in \cite{maggivaldinoci} the authors consider a fractional variant of the classical Gauss free energy for a droplet $E$ confined inside a container $\Omega$,
\[
\mathcal{E}_s(E,\Omega)=I_s(E,E^c\cap\Omega)+\gamma\,I_s(E,\Omega^c)\qquad E\subset\Omega\,,
\]
where $s\in(0,1)$, $\gamma\in(-1,1)$. In \cite[Theorem 1.3, Theorem 1.4]{maggivaldinoci} it was proven that if $E\subset\Omega$ is a volume-constrained minimizer of $\mathcal{E}_s(\cdot,\Omega)$ such that $M=\overline{\partial E\cap\Omega}$ is a $C^{1,\a}$-hypersurface with boundary for some $\sigma\in(s,1)$ and with ${\rm bd}\,(M)\subset\partial\Omega$, then, for $c$ a constant, the Euler-Lagrange equation
\[
\mathbf{H}_E^s(q)+(\gamma-1)\int_{\Omega^c}\frac{1}{|x-q|^{n+s}}dx=c
\]
holds at each $q\in \Omega\cap M$, together with a contact angle condition (fractional Young's law):
\[
\nu_E(x)\cdot\nu_\Omega(x)=\cos(\pi-\theta(s,\gamma))\qquad\forall x\in M\cap\partial\Omega={\rm bd}\,(M)\,,
\]
where $\theta(s,0)=\pi/2$, $\theta(s,(-1)^+)=0$ and $\theta(s,1^-)=\pi$. In the corresponding sessile droplet problem, where one takes $\Omega=H=\{x_n>0\}$, we end up in the situation considered in Corollary \ref{cor axially symmetric}, by setting
\begin{equation}
\label{g mv}
\mathbf{H}_E^s(q)=-c+(1-\gamma)\int_{H^c}\frac{1}{|x-q|^{n+s}}dx\,.
\end{equation}
Because, in this case, $\mathbf{H}_E^s$  is a function of the $q_n$ variable alone, Theorem \ref{thm axially symmetric} implies the axial symmetry of every volume-constrained critical point of the fractional Gauss free energy on a half-space.

\bigskip

\noindent {\bf Acknowledgments:} The author would like to thank Francesco Maggi for mentoring, guidance and helpful discussions. This work was supported by NSF-DMS Grants 1265910 and 1351122.

\section{Blowup of Fractional Mean Curvature}\label{blowup}
We summarize our basic notation and assumptions used in the paper.
\medskip

\noindent\begin{minipage}[t]{0.25\textwidth}
\textbf{Assumption }(h1):
\end{minipage}
\begin{minipage}[t]{0.7\textwidth}
We assume that $s\in(0,1)$, $\a\in(s,1)$, $n\ge2$, and $\mbox{$E\subset\mathbb R^n$}$ is a bounded open connected subset of $H=\{x_n>0\}$ such that \mbox{$M:=\overline{\partial E\cap H}$} is a $C^{2,\a}$-hypersurface with boundary $\mbox{${\rm bd}\,(M):=M\cap\partial H$}$.
\end{minipage}

\medskip

For $q\in M$ we let
\begin{equation*}\label{defAq}
A_qM:=q+T_qM\,.
\end{equation*}
\begin{remark}
The normal vector $\nu_{E}$ on $M\cap H$ extends to $M\cap \partial H.$
\end{remark}
\begin{remark}{\rm
Under assumption (h1), there exist $\eta>0$ and $\gamma>0$ such that, for all $q\in M$, we have
\[
M\cap B_{\eta}(q)\subset P_{\eta,\gamma}\,,
\]
for
\begin{equation}\label{parabola}
P_{\eta,\gamma}(q):=\{x\in B_\eta(q): |x-\mathbf{p}_q x|<\gamma |q-\mathbf{p}_q x|^{2+\a}\}\,,
\end{equation}
where $\mathbf{p}_q x$ is the projection of $x$ onto $A_qM$.}
\end{remark}

From \cite[Lemma 2.1]{ciraolofigallimagginovaga} we know that $\mathbf{H}_E^{s}(q)$ is $C^1$ for all $q\in M\cap H$. But the following proposition shows that, as mentioned in the first section, if $q_n\rightarrow 0$, then $\mathbf{H}_E^{s}(q)$ blows up like $q_n^{-s},$ notably without the assumption that $\nu_E(q)\cdot\nu_H(q)$ is constant.

\begin{proposition} \label{H blow} Let $E,s,\a$ be as in (h1). If $p\in M\cap \partial H,$ $\{q_m\}_{m\in\mathbb N}\subset M\cap H,$ and $q_m\rightarrow p$ as $m\rightarrow\infty$,
\begin{itemize}
\item[(a)]then
\[
\mathbf{H}_E^s(q_m)=\frac{c\big(n,s,\nu_E(p)\cdot \nu_H)\big)}{(q_m\cdot e_n)^s}\Big(1+{\rm O}(1)\Big)\,,
\]
as $m\rightarrow \infty.$
If in addition there is $\sigma\in(-1,1)$ such that $\nu_E(p)\cdot \nu_H=\sigma$ for all $p\in M\cap \partial H$, then
\begin{equation}\label{with sig}
\mathbf{H}_E^s(q)=\frac{c\big(n,s,\sigma\big)}{(q\cdot e_n)^s}\Big(1+{\rm O}(1)\Big)\,,
\end{equation}
as $q\cdot e_n\rightarrow 0,$ for $q\in M\cap H.$
\smallskip
\item[(b)] Moreover
\begin{equation}\label{prop b}
\limsup_{m\rightarrow \infty}\; (q_m\cdot e_n)^{s+1}|\nabla\mathbf{H}_E^s(q_m)|\le c\big(n,s,\nu_E(p)\cdot \nu_H\big)\,.
\end{equation}
\end{itemize}
\end{proposition}
\begin{remark}
{\rm Note that in the proposition the proof of (a) only requires that $M\in C^{1,\a},$ but the proof of (b) requires that $M\in C^{2,\a}.$}\end{remark}
\begin{proof} Consider $q_m$ and $p$ as in the statement. By assumption (h1) there exists $\mbox{$\{p_m\}_{m\in\mathbb N}\subset M\cap \partial H$}$ such that $|p_m-q_m|\rightarrow 0$ as $m\rightarrow \infty$ and $A_{q_m}M\cap\partial H$ is parallel to $A_{p_m}M\cap \partial H$ for each $m$. Let $J_{q_m}$ be the wedge in $H$ bounded by $A_{q_m}M$ and $\partial H$ such that $\nu_{J_{q_m}}(q_m)=\nu_E(q_m).$ Similarly let $J_{p_m}$ be the wedge in $H$ bounded by $A_{p_m}M$ and $\partial H$ such that $\nu_{J_{p_m}}(p_m)=\nu_E(p_m).$ Let $J_{p_m}^*$ be the horizontal translation of $J_{p_m}$ such that $q_m\in J_{p_m}^*,$ see Figure~1. Let $\theta_m$ be defined so that
\[
\cos(\theta_m)=\nu_{J_{p_m}^*}(q_m)\cdot \nu_{J_{q_m}}(q_m)\,.
\]
\begin{figure}
\includegraphics[scale=.55]{wedgeangle}
 \caption{\small{The arrangement of the wedges in the case where $n=2$ and $\nu_E(p_m)\cdot \nu_H$ is negative. Here the shaded area represents the region inside of $E$.}}\label{wedges}
\end{figure} To prove (a), we will compare the mean curvature calculated at $q_m$ for $J_{p_m}^*$ and $J_{q_m}$ to understand $H_E^s(q_m)$ as $m\rightarrow\infty$. We first claim that
\begin{equation}\label{first diff}
|\mathbf{H}_{J_{q_m}}^s(q_m)-\mathbf{H}_E^s(q_m)|\le \frac{K_0(n,s)}{\eta^s}+K_1(\eta,\a,\gamma)\,,
\end{equation}
for a constants $K_0(n,s)>0$ dependent only on $n$ and $s$, $K_1(\eta,\a,\gamma)>0$ dependent only on $\eta, \a$ and $\gamma$, and
for $\eta$ as in \eqref{parabola}. In the following, the constant $K_0(n,s)$ may vary slightly from line to line, but for simplicity we will use the same letter. We use polar coordinates and find that
\begin{align}\label{outside wedge}
\int_{B_{\eta}(q_m)^c}\frac{\tilde{\chi}_{J_{q_m}}(x)}{|x-q_m|^{n+s}}dx
&\le n\omega_n\int_\eta^{\infty}\frac{t^{n-1}}{t^{n+s}}dt\nonumber\\
&= \frac{K_0(n, s)}{\eta^s}\,,
\end{align}
and similarly
\begin{equation}\label{outside E}
\int_{B_{\eta}(q_m)^c}\frac{\tilde{\chi}_{E}(x)}{|x-q_m|^{n+s}}dx\le \frac{K_0(n, s)}{\eta^s}\,.
\end{equation}
By definition of $P_{\eta,\gamma}$ in \eqref{parabola}, we have
\begin{align}\label{in wedge vs E}
{\rm p.v.}\Big|\int_{B_{\eta}(q_m)}\frac{\tilde{\chi}_{J_{q_m}}(x)}{|x-q_m|^{n+s}}-\frac{\tilde{\chi}_{E}(x)}{|x-q_m|^{n+s}}dx\Big|&\le {\rm p.v.}\int_{P_{\eta,\gamma}(q_m)}\frac{dx}{|x-q_m|^{n+s}}\nonumber\\&=K_1(\eta,\a,\gamma)<\infty\,
\end{align}
So by \eqref{outside wedge}, \eqref{outside E}, and \eqref{in wedge vs E} we have \eqref{first diff} as claimed.
We next show that
\begin{equation}\label{second diff}
|\mathbf{H}_{J_{p_m}^*}^s(q_m)-\mathbf{H}_{J_{q_m}}^s(q_m)|\le |\theta_m|\frac{K_0(n,s)}{(q_m\cdot e_n)^s}\,.
\end{equation}
Indeed, we know by symmetry that
\begin{equation}\label{in wedges}
0={\rm p.v.}\int_{B_{q_m\cdot e_n}(q_m)}\frac{\tilde{\chi}_{J_{q_m}}(x)}{|x-q_m|^{n+s}}dz={\rm p.v.}\int_{B_{q_m\cdot e_n}(q_m)}\frac{\tilde{\chi}_{J_{p_m}^*}(z)}{|x-q_m|^{n+s}}dx\,,
\end{equation}
while
\begin{equation}\begin{split}\label{out wedges}
\Big|\int_{B_{q_m\cdot e_n}(q_m)^c}\frac{\tilde{\chi}_{J_{q_m}}(x)}{|x-q_m|^{n+s}}-\frac{\tilde{\chi}_{J_{p_m}^*}(x)}{|x-q_m|^{n+s}}dx\Big|
&\le \int_{q_m\cdot e_n}^{\infty}\frac{|\theta_m|n\omega_nr^{n-1}}{\pi r^{n+s}}dr\\
&=|\theta_m|\frac{K_0(n,s)}{(q_m\cdot e_n)^s}\,.
\end{split}\end{equation}
By \eqref{in wedges} and \eqref{out wedges} we have \eqref{second diff}. Together (\ref{first diff}) and (\ref{second diff}) imply that
\begin{equation}\label{triangle wedge 1}
|\mathbf{H}_{J_{p_m}^*}^s(q_m)-\mathbf{H}^s_E(q_m)|\le \frac{K_0(n,s)}{\eta^s} +K_1(\eta,\a,\gamma)+|\theta_m|\frac{ K_0(n,s)}{(q_m\cdot e_n)^s}\,.
\end{equation}
By scaling (see for example \cite[Theorem 1.4]{maggivaldinoci}) we have
\begin{equation}\label{wedge formula}
\mathbf{H}_{J_{p_m}^*}^s(q_m)=\frac{c(n,s,\nu_{J_{p_m}^*}(q_m)\cdot\nu_H)}{(q_m\cdot e_n)^s}\,,
\end{equation}
where $c(n,s,\cdot)$ is continuous.
Since $|\theta_m|\rightarrow 0$ as $m\rightarrow\infty$ we know $|\nu_{J_{p_m}^*}(q_m)-\nu_{E}(p_m)|\rightarrow 0$ as $m\rightarrow\infty$.  Therefore, because $\nu_{E}(p_m)\rightarrow\nu_E(p)$ as $m\rightarrow\infty,$ \eqref{triangle wedge 1} and \eqref{wedge formula} imply that
\[
\mathbf{H}^s_E(q_m)=\frac{c\big(n,s,\nu_E(p)\cdot\nu_H\big)}{(q_m\cdot e_n)^s}\Big(1+{\rm O}(1)\Big)\,,
\]~as $m\rightarrow\infty.$ Moreover, in the case that $\nu_E(p)\cdot\nu_H=\sigma$ for all $p\in M\cap\partial H$, this implies \eqref{with sig}.

The proof of (b) closely parallels that of part (a), with extra attention to convergence of $\nabla \mathbf{H}_E^s$ throughout. Fix $m\in \mathbb N$ large. We approximate the kernel $|t|^{-(n+s)}$ by $\varphi_{\varepsilon}(t)\in C_c^{\infty}([0,\infty))$ such that $\varphi_\varepsilon\ge 0, \varphi'_\varepsilon\le 0,$ and
\[\left\{
\begin{array}{l}
t^{n+s}\varphi_\varepsilon(t)+t^{n+s+1}|\varphi'_\varepsilon(t)|\le C(n,s)\,,\\
|\varphi'_\varepsilon(t)|\uparrow \frac{n+s}{t^{n+s+1}} \qquad \mbox{ as }\varepsilon\rightarrow 0^+\,,
\end{array}
\right.\quad\forall t> 0\,.
\]
By construction $\varphi_\varepsilon(t)\uparrow |t|^{-(n+s)}$ for all $t>0.$ For $q\in M\cap \partial H$, we set
\[
\mathbf{H}_E^{s,\varepsilon}(q):=\int_{\mathbb R^n}\tilde{\chi}_{E}(x)\varphi_\varepsilon(|x-q|)dx\,,
\] and let $u_{\varepsilon,q}(x)=\varphi_\varepsilon(|x-q|)$.  We have
\begin{equation*}
\nabla \mathbf{H}_E^{s,\varepsilon}(q)=\int_{\mathbb R^n}\tilde{\chi}_E(x)\nabla u_{\varepsilon,q}(x) dx\,.
\end{equation*} Note that, by \cite{ciraolofigallimagginovaga},
\begin{equation}\label{convergence}
\lim_{\varepsilon\rightarrow0}\nabla H_E^{s,\varepsilon}(q)=\nabla H_E^{s}(q)\qquad \forall q\in M\cap H\,.
\end{equation}
To prove \eqref{prop b} it suffices to show that
\begin{equation}\label{real b}
\limsup_{\varepsilon\rightarrow 0}(q_m\cdot e_n)^{s+1}|\nabla H_E^{s,\varepsilon}(q_m)|\le  c\big(n,s,\nu_E(p)\cdot \nu_H\big)\,,
\end{equation}
for large enough $m.$
First we will show that
\begin{equation}\label{derivative triangle I}
\limsup_{\varepsilon\rightarrow 0}|\nabla \mathbf{H}_{J_{q_m}}^{s,\varepsilon}(q_m)-\nabla \mathbf{H}_E^{s,\varepsilon}(q_m)|\le \frac{K(n)}{\eta^{s+1}}+\tilde{K}(\eta,\a,\gamma)\,,
\end{equation}
for some $K(n)$ dependent only on $n$ and $\tilde{K}(\eta,\a,\gamma)>0$ dependent only on $\eta, \a$ and $\gamma.$  In the following, the constant $K(n)$ may vary slightly from line to line, but for simplicity we will use the same letter. We split
\begin{equation}\begin{split}
\label{derivative triangle I 2}
\Big|\int_{\mathbb R^n}(\tilde{\chi}_{E}-\tilde{\chi}_{J_{q_m}})(x)\nabla u_{\varepsilon,q_m}(x)dx\Big|
\le&\Big|\int_{B_{\eta}(q_m)^c}(\tilde{\chi}_{E}-\tilde{\chi}_{J_{q_m}})(x)\nabla u_{\varepsilon,q_m}(x) dx\Big|\\%\nonumber\\
&+\Big|\int_{B_{\eta}(q_m)}(\tilde{\chi}_{E}-\tilde{\chi}_{J_{q_m}})(x)\nabla u_{\varepsilon,q_m}(x) dx\Big|\,.\qquad
\end{split}\end{equation}
By applying the monotone convergence theorem in $\varepsilon$ and by using polar coordinates we see that there is a constant $K(n)$ dependent only on $n$ such that
\begin{align}\label{outside wedge dir}
\limsup_{\varepsilon\rightarrow0}\Big|\int_{B_{\eta}(q_m)^c}\tilde{\chi}_{J_{q_m}}(x)\nabla u_{\varepsilon,q_m}(x) \;dx\Big|&\le
\limsup_{\varepsilon\rightarrow0}\int_{B_{\eta}(q_m)^c}|\nabla u_{\varepsilon,q_m}(x)| \;dx\nonumber\\
&\le K(n)\int_\eta^{\infty}\frac{t^{n-1}}{t^{n+s+1}} \; dt\nonumber\\
&=\frac{ K(n)}{\eta^{s+1}}\,,
\end{align}
and similarly
\begin{equation}\label{outside E dir}
\limsup_{\varepsilon\rightarrow 0}\Big|\int_{B_{\eta}(q_m)^c}\tilde{\chi}_{E}(x)\nabla u_{\varepsilon,q_m}(x)dx\Big|\le\frac{K(n)}{\eta^{s+1}}\,.
\end{equation}
By definition of $P_{\eta,\gamma}$ in \eqref{parabola}, we have
\begin{align}\label{pinchie}
 \Big|\int_{B_{\eta}(q_m)}(\tilde{\chi}_{E}-\tilde{\chi}_{J_{q_m}})(x)\nabla u_{\varepsilon,q_m}(x) dx\Big|
&\le\Big|\int_{P_{\eta,\gamma}(q_m)}(\tilde{\chi}_{E}-\tilde{\chi}_{J_{q_m}})(x)\nabla u_{\varepsilon,q_m}(x) dx\Big|\nonumber\\
&\le\int_{P_{\eta,\gamma}(q_m)}2|\varphi'_\varepsilon(|x-q_m|)| dx\,.
\end{align}
Therefore, because
\[
 \lim\limits_{\varepsilon\rightarrow 0}\int_{P_{\eta,\gamma}(q_m)}2|\varphi'_\varepsilon(|x-q_m|)|\; dx= {\rm p.v.}\int_{P_{\eta,\gamma}(q_m)}\frac{2dx}{|x-q_m|^{n+s+1}}=\tilde{K}(\eta,\a,\gamma)<\infty\,,
\]
and by the monotone convergence theorem in $\varepsilon$, we can plug \eqref{outside wedge dir}, \eqref{outside E dir} and \eqref{pinchie} into \eqref{derivative triangle I 2} and get \eqref{derivative triangle I}.
Next we prove that
\begin{equation}\label{second diff II}
\limsup_{\varepsilon\rightarrow0}|\nabla\mathbf{H}_{J_{p_m}^*}^{s,\varepsilon}(q_m)-\nabla\mathbf{H}_{J_{q_m}}^{s,\varepsilon}(q_m)|\le |\theta_m|\frac{K(n)}{(q_m\cdot e_n)^{s+1}}\,.
\end{equation}
We know, by symmetry, that, for any $\varepsilon$,
\begin{equation}\label{in wedges diff}
0=\int_{B_{q_m\cdot e_n}(q_m)}\tilde{\chi}_{J_{q_m}}(x)\nabla u_{\varepsilon,q_m}(x) \;dx=\int_{B_{q_m\cdot e_n}(q_m)}\tilde{\chi}_{J_{p_m}^*}(x)\nabla u_{\varepsilon,q_m}(x) \;dx\,,
\end{equation}
so,
\begin{align}
\Big|\int_{\mathbb R^n}(\tilde{\chi}_{J_{q_m}}-\tilde{\chi}_{J_{p_m}^*})(x)\nabla u_\varepsilon(x)dx\Big|
&=\Big|\int_{B_{q_m\cdot e_n}(q_m)^c}(\tilde{\chi}_{J_{q_m}}-\tilde{\chi}_{J_{p_m}^*})(x)\nabla u_\varepsilon(x)dx\Big|\nonumber\\
&\le\int_{B_{q_m\cdot e_n}(q_m)^c}|\tilde{\chi}_{J_{q_m}}-\tilde{\chi}_{J_{p_m}^*}|(x)\big|\varphi'_\varepsilon(|x-q_m|)\big|dx\,.\nonumber
\end{align}
Therefore, by monotone convergence theorem in $\varepsilon$, there is a constant $K(n)$ such that
\begin{equation}\label{out wedges dif}
\limsup_{\varepsilon\rightarrow 0}\Big|\int_{\mathbb R^n}(\tilde{\chi}_{J_{q_m}}-\tilde{\chi}_{J_{p_m}^*})(x)\nabla u_\varepsilon(x)dx\Big|     \le|\theta_m|\frac{K(n)}{(q_m\cdot e_n)^{s+1}}\,,
\end{equation}
which implies \eqref{second diff II}. Together, \eqref{derivative triangle I} and \eqref{second diff II} imply that
\begin{equation}\label{triangle wedge}
\limsup_{\varepsilon\rightarrow0}|\nabla \mathbf{H}_{J_{p_m}^*}^{s,\varepsilon}(q_m)\cdot \t_m-\nabla \mathbf{H}^{s,\varepsilon}_E(q_m)\cdot \t_m|\le \frac{K(n)}{\eta^{s+1}}+\tilde{K}(\eta,\a,\gamma)+|\theta_m|\frac{K(n)}{(q_m\cdot e_n)^{s+1}}\,.
\end{equation}
For $m$ large we use \eqref{convergence}, exploit symmetry as in \eqref{in wedges diff}, apply monotone convergence theorem in $\varepsilon$, and use polar coordinates as in \eqref{outside wedge dir}, to find that there is a constant $c(n,s,\nu_E(p_m)\cdot\nu_H )>0$ dependent on $n,$ $s$, and $\nu_E(p_m)\cdot\nu_H$, such that
\begin{align}\label{dif wedge formula}
|\nabla\mathbf{H}_{J_{p_m}^*}^s(q_m)|&=\lim\limits_{\varepsilon\rightarrow 0}\Big|\int_{\mathbb R^n}\tilde{\chi}_{J_{p_m}^*}(x)\nabla u_{\varepsilon,q_m}(x) dx\Big|\nonumber\\
&=\lim\limits_{\varepsilon\rightarrow 0}\Big|\int_{B_{q_m\cdot e_n}(q_m)^c}\tilde{\chi}_{J_{p_m}^*}(x)\nabla u_{\varepsilon,q_m}(x) dx\Big|\nonumber\\
&\le \int_{B_{q_m\cdot e_n}(q_m)^c}\frac{(n+s)|\tilde{\chi}_{J_{p_m}^*}(x)|}{|x-q_m|^{n+s+1}}dx\nonumber\\
&\le\frac{ c\big(n,s,\nu_E(p_m)\cdot\nu_H\big)}{(q_m\cdot e_n)^{s+1}}\,,
\end{align}
as $m\rightarrow \infty,$ where $c(n,s,\cdot)$ is continuous.
Letting $m\rightarrow\infty$ in \eqref{triangle wedge} and \eqref{dif wedge formula}, by $|\theta_m|\rightarrow 0$ and $\nu_{E}(p_m)\rightarrow\nu_E(p)$ as $m\rightarrow\infty$, we have
\[
\limsup_{m\rightarrow \infty}(q_m\cdot e_n)^{s+1}\mathbf{H}^s_E(q_m)\le c\big(n,s,\nu_E(p)\cdot\nu_H\big)\,.
\]
\end{proof}

\section{Proof of Almost Axial Symmetry}

In this section we prove part (a) of Theorem \ref{thm axially symmetric}. Recall that in addition to assumption (h1) we assume the existence of $\sigma\in(-1,1)$ such that
\begin{equation*}
\nu_E(q)\cdot\nu_H(q)=\sigma\qquad\forall q\in M\cap\partial H\,.
\end{equation*}

\begin{proof}[Proof of Theorem 1.1 (a)]  We start by introducing notation used in the proof. Let  $e\in S^{n-1}\cap\partial H$. Without loss of generality we assume that $e=e_1.$ We define
\[
\begin{array}{rlll}
\pi_\mu&=&\{x_1=\mu\}& \, \mbox{a hyperplane perpendicular to $e_1$}\,,\\
\rho_\mu(x)&=&(2\lambda-x_1,...,x_n)&\, \mbox{the reflection of $x$ across $\pi_\mu$}\,,
\end{array}
\]
and we set
\begin{equation}\label{lambda}
\lambda=\inf\big\{\mu\in\mathbb R:\rho_\mu(E)\cap \{x_1< \mu\}\subset E \big\}
\end{equation}
to be the critical value for $\mu$ for the moving planes argument. By regularity of the boundary, we know $\lambda$ is well defined. We call $\pi_{\lambda}$ the critical hyperplane and, as long as there is no confusion, we will denote
\[
\rho(x)=\rho_{\lambda}(x)\,.
\]
As in \cite{Wente80}, at least one of the four following cases holds in the critical position (see Figure~2):
\begin{MYitemize}
\item\textit{ Case one:} $\partial \rho(E)$ is tangent to $\partial E$ at some point $p\in M\cap \pi_{\lambda}^c\cap{\rm bd}\,(M)^c\,$.

\item\textit{ Case two:} $\partial \rho(E)$ is tangent to $\partial E$ at some point $p\in M\cap\pi_{\lambda}\cap{\rm bd}\,(M)^c\,$.

\item\textit{ Case three:} $\partial \rho(E)$ is tangent to $\partial E$ at some point $p\in M\cap\pi_{\lambda}^c\cap{\rm bd}\,(M)\,$.

\item\textit{ Case four:} $\partial \rho(E)$ is tangent to $\partial E$ at some point $p\in M\cap\pi_{\lambda}\cap{\rm bd}\,(M)\,$.
\end{MYitemize}

\begin{figure}
\includegraphics[scale=.45]{cases2}
 \caption{\small{The four cases of the moving planes argument.}}\label{cases}
\end{figure}
\smallskip

To prove almost symmetry our goal is to show that $|E\Delta \rho(E)|$ is bounded above by a multiple of $\sqrt{\delta_s(E)}$ in each case. In particular we will prove that
\begin{equation}\label{int diff}
\int_{E\Delta \rho(E)}\hbox{dist}(x,\pi_\lambda)dx\le \frac{5}{2(n+s)}\mathrm{diam}(E)^{n+1}\delta_s(E)\,.
\end{equation}
Case one and case two follow \cite{ciraolofigallimagginovaga}, but we include the proofs for the sake of completeness.

\medskip
\noindent\textit{Case one}:
Suppose there is $p\in\{x_1<\lambda\}$ such that $p\in M\cap \rho(M)\cap H.$ Then
\begin{equation*}
\begin{split}
\mathbf{H}_E^s(\rho(p))-\mathbf{H}_E^s(p)&=\mathbf{H}_{\rho(E)}^{s}(p)-\mathbf{H}_E^{s}(p)\\%By sim and bc $ H^c$ same for both
&=2\Big( \int_{E\setminus \rho(E)}\frac{1}{|x-p|^{n+s}}dx-\int_{\rho(E)\setminus E}\frac{1}{|x-p|^{n+s}}dx \Big)\\%sym and all thats left see if expand H
&=2\int_{E\setminus \rho(E)}\Big(\frac{1}{|x-p|^{n+s}}-\frac{1}{|\rho(x)-p|^{n+s}}\Big)dx \,
\end{split}
\end{equation*}
Denote $x=(x_1,\bar{x}).$ By definition $\rho(x)=(2\lambda-x_1,\bar{x}),$ so
\begin{equation}\begin{split}\label{case one calc two}
\Big(\frac{|\rho(x)-p|}{|x-p|}\Big)^{2}&=\frac{(2\lambda-2p_1-(x_1-p_1))^2+(\bar{x}-\bar{p})^2}{|x-p|^2}\\
&=1+ \frac{(2\lambda-2p_1)^2-2(2\lambda-2p_1)(x_1-p_1)}{|x-p|^2}\\
&=1+ \frac{4(\lambda-p_1)(\lambda-x_1)}{|x-p|^2}\,,
\end{split}\end{equation}
and thus
\begin{equation}\label{case 1 mid}\begin{split}
\frac{1}{|x-p|^{n+s}}-\frac{1}{|\rho(x)-p|^{n+s}}&=\frac{1}{|\rho(x)-p|^{n+s}}\Big[\Big(\frac{|\rho(x)-p|}{|x-p|}\Big)^{n+s}-1\Big]\\
&=\frac{1}{|\rho(x)-p|^{n+s}}\Big[\Big(1+\frac{4(\lambda-x_1)(\lambda-p_1)}{|x-p|^2}\Big)^{\frac{n+s}{2}}-1\Big]\,
\end{split}
\end{equation}
Therefore, by the convexity of $f(t)=(1+t)^{(n+s)/2}-1$, we know that
\begin{equation}\label{case one pos}
\frac{1}{|x-p|^{n+s}}-\frac{1}{|\rho(x)-p|^{n+s}}\ge\frac{2(n+s)(\lambda-x_1)(\lambda-p_1)}{|\rho(x)-p|^{n+s}|x-p|^2}\ge\frac{2(n+s)(\lambda-x_1)(\lambda-p_1)}{\mathrm{diam}(E)^{n+s+2}}\ge0\,,
\end{equation}
for $x\in E\setminus \rho(E)$. Indeed, for $x\in E\setminus \rho(E),$ we have $|x-p|=|\rho(x)-p|\le\mathrm{diam}(E)$, $\lambda-x_1\ge0$, and $|p-\rho(p)|=2(\lambda-p_1)>0$.
So
\begin{equation*}\begin{split}
\frac{\delta_s(E)}{\mathrm{diam}(E)^{s+1}}&\ge\frac{|\mathbf{H}_E^s(\rho(p))-\mathbf{H}_E^s(p)|}{|\rho(p)-p|}= \frac{|\mathbf{H}_E^s(\rho(p))-\mathbf{H}_E^s(p)|}{2(\lambda-p_1)}\\
&\ge\frac{2(n+s)}{\mathrm{diam}(E)^{n+s+2}}\int_{E\setminus \rho(E)}|x_1-\lambda |\,,
\end{split}\end{equation*}
which implies
\[
\int_{E\Delta \rho(E)}|x_1-\lambda |\le\frac{1}{(n+s)} \mathrm{diam}(E)^{n+1}\delta_s(E)\,,
\]
and therefore \eqref{int diff} in case one.

\medskip

\noindent\textit{Case two:} Suppose that there is $p\in\{x_1=\lambda\}\cap H$ such that $M$ is orthogonal to $\pi_\lambda$. As in the proof of Proposition \ref{H blow}, we approximate the kernel $|t|^{-(n+s)}$ by $\varphi_{\varepsilon}(t)\in C_c^{\infty}([0,\infty))$ such that $\varphi_\varepsilon\ge 0, \varphi'_\varepsilon\le 0,$ and
\[\left\{
\begin{array}{l}
t^{n+s}\varphi_\varepsilon(t)+t^{n+s+1}|\varphi'_\varepsilon(t)|\le C(n,s)\,,\\
|\varphi'_\varepsilon(t)|\uparrow \frac{n+s}{t^{n+s+1}} \qquad \mbox{ as }\varepsilon\rightarrow 0^+\,,
\end{array}
\right.\quad\forall t> 0\,.
\]
Then, by construction, $\varphi_\varepsilon(t)\uparrow |t|^{-(n+s)}$ for all $t>0.$ Recall that for any set $F\in \mathbb R^n$ we set $\tilde{\chi}_F(x)=\chi_{F^c}(x)-\chi_F(x).$ We define
\[
\mathbf{H}_E^{s,\varepsilon}(q):=\int_{\mathbb R^n}\tilde{\chi}_{E}(x)\varphi_\varepsilon(|x-q|)dx\,,
\] and let $u_\varepsilon(x)=\varphi_\varepsilon(|x-p|)$.
For all $a>0$, we know, by \cite{ciraolofigallimagginovaga}, that $\mathbf{H}_E^{s,\varepsilon}\rightarrow\mathbf{H}_E^{s}$ in $C^{1}(\partial E\cap \{x_n>a\})$ as $\varepsilon\rightarrow 0$. So we have
\begin{equation*}
\lim\limits_{\varepsilon\rightarrow 0}\nabla \mathbf{H}_E^{s,\varepsilon}(p)\cdot e_1=\lim\limits_{\varepsilon\rightarrow 0}\int_{\mathbb R^n}\tilde{\chi}_E(x)\nabla u_\varepsilon(x)\cdot e_1dx\,.
\end{equation*}
Because $\nabla u_\varepsilon(x)\cdot e_1=\varphi'_\varepsilon(|x-p|)\frac{(x-p)\cdot e_1}{|x-p|}$ is odd with respect to the hyperplane $\{x_1=\lambda \}$, we know that
\[
\int_{ H^c}\nabla u_\varepsilon(x)\cdot e_1dx=0, \quad \int_{E\cap \rho(E)}\nabla u_\varepsilon(x)\cdot e_1dx=0, \quad \text{and } \quad \int_{(E\cup \rho(E))^c\cap H}\nabla u_\varepsilon(x)\cdot e_1dx=0\,.
\]
Therefore
\begin{equation*}
\begin{split}
\nabla \mathbf{H}_E^{s,\varepsilon}(p)\cdot e_1&=\int_{\mathbb R^n}[\chi_{E^c}-\chi_{E}](x)\nabla u_\varepsilon(x)\cdot e_1dx\\
&=\int_{\mathbb R^n}\Big[\chi_{H^c}+\chi_{(E\cup \rho(E))^c\cap H}+\chi_{\rho(E)\setminus E}-\chi_{E\setminus \rho(E)}-\chi_{E\cap \rho(E)}\Big](x)\nabla u_\varepsilon(x)\cdot e_1dx\\
&=-2\int_{E\setminus \rho(E)}\nabla u_\varepsilon(x)\cdot e_1dx\,.
\end{split}
\end{equation*}
So, by the negativity of $\frac{(x-p)\cdot e_1}{|x-p|}$ for $x\in E\setminus \rho(E)\subset \{x_1<\lambda \}$ and by the monotone convergence of $|\varphi_\varepsilon'(t)|$ in $\varepsilon$, we have
\begin{equation*}
\nabla \mathbf{H}_E^{s}(p)\cdot e_1=-2(n+s)\int_{E\setminus \rho(E)}\frac{(p-x)\cdot e_1}{|x-p|^{n+s+2}}dx\,.
\end{equation*}
Therefore, because $|\nabla \mathbf{H}_E^s(p)\cdot e_1|\le \mathrm{diam}(E)^{-(s+1)}\delta_s(E)$, and by
\[
\frac{(p-x)\cdot e_1}{|x-p|^{n+s+2}}\ge \frac{|x_1-\lambda |}{\mathrm{diam}(E)^{n+s+2}}\qquad \forall x\in E\setminus\rho(E)\,,
\]
we have that
\[
\frac{\delta_s(E)}{\mathrm{diam}(E)^{s+1}}\ge\frac{2(n+s)}{\mathrm{diam}(E)^{n+s+2}}\int_{E\setminus \rho(E)}|x_1-\lambda |dx\,.
\]

\medskip

\noindent\textit{Case three}: We now consider the case where $p\in{\rm bd}\,(M)\cap\{x_1<\lambda\}$. As in case one we would like to consider the difference
\[
\mathbf{H}^s_{\rho(E)}(p)-\mathbf{H}^s_E(p)\,,
\]
but both terms equal infinity for $p\in{\rm bd}\,(M)$. So we use an approximation argument to improve our understanding of this difference as a limit. A key tool for this approximation, and also for a similar approximation in case four, will be the local graphicality of $M$ and $\rho(M)$ over $A_pM=A_p\rho(M)$ around $p.$ (Recall from Section \ref{blowup} that $A_qM:=q+T_qM$ for $q\in M$.) Let $U\subset A_p\cap H$ be a local neighborhood of $p$ in $A_pM\cap H$ such that we can define $v: U\rightarrow M$ to be the normal map along $\nu_E(p)$ to $M$. There is a set $U^*\subset U$ such that $v^*: U^* \rightarrow\rho( M)$ can be defined as the normal map along $\nu_E(p)$ to $\rho(M)$. If $U^*\ne U$ then we can reset it so that the sets are equal, because $U^*$ contains a neighborhood of $p$ in $A_pM$  on which $v$ may be defined. We can expand $v$ and $v^*$ as
\[
v(\hat q)=v(p)+\nabla v(p)[\hat{q}-p] +O(|\hat{q}-p|^{2})=p+\nabla v(p)[\hat{q}-p] +O(|\hat{q}-p|^{2})
\]
and
\[
v^*(\hat q)=v^*(p)+\nabla v^*(p)[\hat{q}-p]+O(|\hat{q}-p|^{2})=p+\nabla v^*(p)[\hat{q}-p]+O(|\hat{q}-p|^{2})\,,
\]
for $\hat{q}\in A_pM\cap H$ with $|\hat{q}-p|$ small enough.  Let
\[
\hat{q}_m:=p-\frac{\nu_{co}^M(p)}{m} \qquad\mbox{for $m\in\mathbb N$}\,,
\]
where $\nu_{co}^M(p)$ is the conormal vector for $p$ with respect to $M.$ Note that this conormal vector is also the same for $\rho(M)$ because $p$ is a point of tangency between $M$ and $\rho(M)$, $M$ is $C^{2,\a}$, and $M$ has constant contact angle with $\partial H.$
Let $q_m=v(\hat{q}_m)$ and $q_m^*=v^*(\hat{q}_m)$. We can now consider the correct analogue of $\mathbf{H}^s_{\rho(E)}(p)-\mathbf{H}^s_E(p)$ to be the limit of
\begin{equation}\label{case 3 diff}
Q_m:=\mathbf{H}_{\rho(E)}^{s}(q_m^*)-\mathbf{H}_E^{s}(q_m)\,,
\end{equation}
as $m\rightarrow\infty.$

\smallskip

  \noindent {\it Step one}: We show that
  \begin{equation}\label{*}
  \limsup_{m\rightarrow\infty}|Q_m|\le  \frac{5}{2}|\lambda-p_1|\frac{\delta_s(E)}{\mathrm{diam}(E)^{s+1}}
  \end{equation}
 as $m\rightarrow \infty$. The claim \eqref{*} is trivial if $\sigma= 0$ because then $q_m$ and $q_m^*$ have the same vertical component. Indeed, by definition of $\delta_s$, we then have
 \[
| Q_m|\le \frac{\delta_s(E)}{\mathrm{diam} (E)^{s+1}}|q_m-\rho(q_m)|\,,
 \]
which implies \eqref{*} because $|q_m-\rho(q_m)|\rightarrow |p-\rho(p)|=2|p_1-\lambda|$ as $m\rightarrow \infty$.
 Otherwise, let
 \[
 e_p:=\frac{\nu_E(p)-(\nu_E(p)\cdot e_n)e_n}{|\nu_E(p)-(\nu_E(p)\cdot e_n)e_n|}\,.
 \]
 For $m$ large enough, and for each $q_m^*\in \rho(M)$ close enough to $p$, let $\tilde{q}_m\in M$ be defined to be the unique projection of $q_m^*$ onto $M$ along $e_p.$ Note that the definition of $\tilde{q}_m$ enforces that $q_m,q_m^*,\tilde{q}_m,$ and $\hat{q}_m$ are all contained in the plane $T:=\mathrm{span}\{\nu_E(p),\nu_{co}^M(p)\}$, and by the continuity of the projection we have that $\tilde{q}_m\rightarrow p$ as $m\rightarrow\infty.$

 By the triangle inequality
\begin{equation}\label{**}
|Q_m|=\big|\mathbf{H}_{\rho(E)}^{s}(q_m^*)-\mathbf{H}_E^{s}(q_m)\big|\le\big|\mathbf{H}_{\rho(E)}^{s}(q_m^*)-\mathbf{H}_{E}^{s}(\tilde{q}_m)\big|+\big|\mathbf{H}_{E}^{s}(\tilde{q}_m)-\mathbf{H}_E^{s}(q_m)\big|\,.
\end{equation}
By definition of $\delta_s(E)$, we know
\[
\big|\mathbf{H}_{\rho(E)}^{s}(q_m^*)-\mathbf{H}_{E}^{s}(\tilde{q}_m)\big|=\big|\mathbf{H}_E^{s}(\rho(q_m^*))-\mathbf{H}_{E}^{s}(\tilde{q}_m)\big|\le |\rho(q_m^*)-\tilde{q}_m|\frac{\delta_s(E)}{\mathrm{diam}(E)^{s+1}}\,,
\]
for any given $m$. But as $m\rightarrow \infty$ we know $ |\rho(q_m^*)-\tilde{q}_m|\rightarrow |\rho(p)-p|=2|\lambda-p_1|$. So for $m$ large enough
\[
\big|\mathbf{H}_{\rho(E)}^{s}(q_m^*)-\mathbf{H}_{E}^{s}(\tilde{q}_m)\big|\le \frac{5}{2}|\lambda-p_1|\frac{\delta_s(E)}{\mathrm{diam}(E)^{s+1}}.
\]
Therefore to prove \eqref{*} it suffices to show that
\begin{equation}\label{*2}
\limsup_{m\rightarrow 0}|\mathbf{H}_{E}^{s}(\tilde{q}_m)-\mathbf{H}_E^{s}(q_m)|=0\,.
\end{equation}
By Proposition \ref{H blow}(b), there is a constant $c= c\big(n,s,\sigma\big)>0$ such that, if $h_m=\min\{q_m\cdot e_n, \tilde{q}_m\cdot e_n\}$, then
\begin{equation}\label{unif dif apl}
|\mathbf{H}_{E}^{s}(\tilde{q}_m)-\mathbf{H}_E^{s}(q_m)|\le c\big(n,s,\sigma\big)\frac{  |q_m-\tilde{q}_m|}{h_m^{s+1}}\,.
\end{equation}
We claim that
\begin{equation}\label{case three step one}
\lim_{m\rightarrow\infty}\frac{|q_m-\tilde{q}_m|}{|h_m|^{1+s}}=0\,,
\end{equation}
which would suffice to prove \eqref{*2}.
 Because $\tilde{q}_m\in M$, for $m$ large enough there is $\hat{t}_m\in A_pM$ such that $v(\hat{t}_m)=\tilde{q}_m.$ Fix $m\in\mathbb N$ large. Let  $B_m=\max\{|\hat{q}_m-p|,|\hat{t}_m-p|\}$ and $\mbox{$b_m=\min\{|\hat{q}_m-p|,|\hat{t}_m-p|\}$}.$ So $B_m-b_m=|\hat{t}_m-\hat{q}_m|$.
 To prove \eqref{case three step one} we will show that there exist constants $C(\sigma), C_1(\sigma)>0$, dependent only on $\sigma,$ such that
   \begin{equation}\label{numerator}
 |q_m-\tilde{q}_m|\le C(\sigma)B_m^{2+\a}\,,
 \end{equation}
 and
 \begin{equation}\label{denominator}
 h_m\ge C_1(\sigma)\Big(B_m-C(\sigma)B_m^{2+\a}\Big)\,.
 \end{equation}
Together \eqref{numerator} and \eqref{denominator} give
 \begin{equation*}\begin{split}
 \limsup_{m\rightarrow\infty}\frac{|q_m-\tilde{q}_m|}{|h_m|^{1+s}}&\le \lim_{m\rightarrow\infty} \frac{C(\sigma)B_m^{2+\a}}{C_1(\sigma)(B_m-C(\sigma)B_m^{2+\a})^{s+1}}\\&= \lim_{m\rightarrow\infty}\frac{C(\sigma)B_m^{2+\a}}{C_1(\sigma)B_m^{1+s}(1-C(\sigma)B_m^{1+\a})^{1+s}}=0\,,
 \end{split}\end{equation*}
 which implies \eqref{case three step one} because $B_m\rightarrow 0$ as $m\rightarrow \infty$.

To prove  \eqref{numerator} and \eqref{denominator} we will exploit local graphicality of $M$ and $\rho(M)$ over $A_pM$, considered in $T$. In this plane, in a neighborhood of $p,$ there is an ordering of $M$, $\rho(M)$, and $A_pM$ with respect to $e_p.$ In particular, if we let $\hat{a}_m$ be the projection of $q_m^*$ along $e_p$ to $A_pM$, then $\tilde{q}_m\cdot e_p $, $q_m^*\cdot e_p,$ and $\hat{a}_m\cdot e_p$ have a fixed order for large enough $m$. Because the moving planes method forces $\tilde{q}_m\cdot e_p \ge q_m^*\cdot e_p,$ there are three possible orderings of these coordinates. Combined with the fact that we can consider $\sigma>0$ or $\sigma<0$, we have six total subcases to consider, see Figure 3. Because each case takes place in $T$ we set up the following notation. Define
\[
f(d):=v(p-d\nu_{co}^M(p))\cdot\big(\nu_E(p)\big)\,,
\qquad\mbox{ and }\qquad
f^*(d):=v^*(p-d\nu_{co}^M(p))\cdot\big(\nu_E(p)\big)\,.
\]
Because $v$ and $v^*$ are $C^{2,\a}$, with $v(p)=v^*(p)=0$, and with $\nabla v(p)\cdot\nu_E(p)=\nabla v^*(p)\cdot\nu_E(p)=0$, there exists a constant $\gamma>0$ such that, $f$ and $f^*$ are monotone, and
\begin{equation}\label{f reg}
|f(d)|,|f^*(d)|\le \gamma d^{2+\a},
\end{equation}
for $d$ small enough. Moreover for $m$ large enough, there is a constant $c\le1$
such that
\begin{equation}\label{lip bound f}
|f(d_2)-f(d_1)|\le c|d_2-d_1|\qquad \forall d_1,d_2\mbox{ small enough}.
\end{equation}
Let $\tilde{f}(d):=f(d)+f^*(d).$
We first prove \eqref{numerator} and then \eqref{denominator}. By \eqref{lip bound f}, we have
\begin{equation}\label{real num}
 |q_m-\tilde{q}_m|=\sqrt{(B_m-b_m)^2+(f(B_m)-f(b_m))^2}\le 2|B_m-b_m|\,,
\end{equation}
so we only need to prove there is a constant $C(\sigma)$ dependent only on $\sigma$ such that
\begin{equation}\label{real nume bound}
|B_m-b_m|\le C(\sigma)\tilde{f}(B_m) \le C(\sigma)B_m^{2+\a}\,,
\end{equation}
and we have \eqref{numerator}. In each subcase the proof comes from considering one or two right triangles in $T$ with one edge parallel to $A_pM$, one edge perpendicular to $A_pM$, and one edge parallel to $\partial H,$ where the last edge is contained in or equal to the line segment connecting $q_m^*$ and $\tilde{q}_m$. Define $\theta$ so that
\[
\cos(\theta)=\nu_E(p)\cdot\nu_H=\sigma\,.
\]
  In the first three cases, where $\sigma>0$, note that $\theta<\pi/2$, and for the last three, where $\sigma<0$, note that $\theta>\pi/2$.

\begin{figure}
\includegraphics[scale=.68]{case32a}
 \caption{\small{The six subcases used in the proof of \eqref{numerator}. We consider cases depending on the sign of $\sigma$ and on the position of $M$ and $\rho(M)$ with respect to $A_pM$.} Here the shaded area represents the region inside of $E$.}\label{six cases}
\end{figure}

\noindent \textit{Subcase one}: Suppose $\sigma>0$ and $\hat{a}_m\cdot e_p \le q_m^*\cdot e_p\le \tilde{q}_m\cdot e_p$.
Then
\[
|B_m-b_m|\le \cot(\theta)f(B_m) \le C(\sigma)\tilde{f}(B_m)\,.
\]

\noindent \textit{Subcase two}: Suppose $\sigma>0$ and $q_m^*\cdot e_p\le \hat{a}_m\cdot e_p \le \tilde{q}_m\cdot e_p$. Then
\[
|B_m-b_m|\le \cot(\theta)(f(B_m)+f^*(b_m)) \le C(\sigma)\tilde{f}(B_m)\,.
\]

\noindent \textit{Subcase three}: Suppose $\sigma>0$ and $q_m^*\cdot e_p\le\tilde{q}_m\cdot e_p \le  \hat{a}_m\cdot e_p$. Then
\[
|B_m-b_m|\le \cot(\theta)(f^*(b_m)) \le C(\sigma)\tilde{f}(B_m)\,.
\]

\noindent \textit{Subcase four}: Suppose $\sigma<0$ and $\hat{a}_m\cdot e_p \le q_m^*\cdot e_p\le \tilde{q}_m\cdot e_p$. Then
\[
|B_m-b_m|\le \cot(\pi-\theta)f(b_m) =-\cot(\theta)f(b_m) \le C(\sigma)\tilde{f}(B_m)\,.
\]

\noindent \textit{Subcase five}: Suppose $\sigma<0$ and $q_m^*\cdot e_p\le \hat{a}_m\cdot e_p \le \tilde{q}_m\cdot e_p$. Then
\[
|B_m-b_m|\le -\cot(\theta)(f(b_m)+f^*(B_m)) \le C(\sigma)\tilde{f}(B_m)\,.
\]

\noindent \textit{Subcase six}: Suppose $\sigma<0$ and $q_m^*\cdot e_p\le\tilde{q}_m\cdot e_p \le  \hat{a}_m\cdot e_p$. Then
\begin{equation*}
|B_m-b_m|\le -\cot(\theta)(f^*(B_m)) \le C(\sigma)\tilde{f}(B_m)\,.
\end{equation*}

\begin{figure}
\includegraphics[scale=.68]{case31b2}
 \caption{\small{The four subcases used to prove \eqref{denominator}. We can reduce the number of cases because we only need to consider the point $q_m$ or $\tilde{q}_m$ that minimizes the distance to $\partial H$. Here the shaded area represents the region inside of $E$.}}\label{six cases 2}
\end{figure}

Thus, by \eqref{f reg}, we have completed the proof of \eqref{numerator}. We now proceed with the proof of \eqref{denominator}. We will see that we can combine subcases one and two as well as subcases four and five, see Figure 4. Let $\bar{q}_m=q_m$ and $\hat{s}_m=\hat{q}_m$ if $q_m\cdot e_n=h_m$, and $\bar{q}_m=\tilde{q}_m$ and $\hat{s}_m=\hat{t}_m$  otherwise. Note that $b_m=|\hat{s}_m-p|.$  Exploiting that $h_m$ is approximately equal to $\hat{s}_m\cdot e_n$ will be essential in proving \eqref{denominator}.

\noindent \textit{Subcases one and two }: Suppose $\sigma>0$ and $\hat{a}_m\cdot e_p \le  \tilde{q}_m\cdot e_1$.
Then, by \eqref{f reg}, for $m$ large enough
\[
h_m= b_m\sin\theta-f(b_m)\sin(\pi/2-\theta)=b_m\sin(\theta)-f(b_m)\cos\theta\ge b_m\sin(\theta)/2\,.
\]

\noindent \textit{Subcase three}: Suppose $\sigma>0$ and $   \tilde{q}_m\cdot e_p\le\hat{a}_m\cdot e_p$. Then
\[
h_m= b_m\sin\theta+f(b_m)\sin(\pi/2-\theta)= b_m\sin(\theta)+f(b_m)\cos\theta\ge b_m\sin(\theta)\,.
\]

\noindent \textit{Subcases four and five}: Suppose $\sigma<0$ and $\hat{a}_m\cdot e_p \le  \tilde{q}_m\cdot e_p$.
 Then
\[
h_m= b_m\sin(\pi-\theta)+f(b_m)\sin(\pi/2-(\pi-\theta))= b_m\sin(\theta)+f(b_m)|\cos\theta|\ge b_m\sin(\theta)\,.
\]

\noindent \textit{Subcase six}: Suppose $\sigma<0$ and $\tilde{q}_m\cdot e_p\le\hat{a}_m\cdot e_p$.Then, by \eqref{f reg}, for $m$ large enough
\[
h_m= b_m\sin(\pi-\theta)-f(b_m)\sin(\pi/2-(\pi-\theta))= b_m\sin(\theta)-f(b_m)|\cos\theta|\ge b_m\sin(\theta)/2\,.
\]
Thus in each case we conclude that $h_m\ge b_m\sin(\theta)/2$. Because \eqref{real nume bound} implies
\[
 b_m=B_m-(B_m-b_m)\ge B_m-C(\sigma)B_m^{2+\a}\,,
\]we therefore have \eqref{denominator}, which completes the proof of step one.

\smallskip

\noindent {\it Step two}: In this step we will define an approximation $Q_m^\varepsilon$ of $Q_m$ and show that there is $\varepsilon(m)$ defined so that
\begin{equation}\label{Step Two}
\liminf\limits_{m\rightarrow \infty}Q_m^{\varepsilon(m)}\ge\int_{E\setminus \rho(E)}\frac{2(n+s)(\lambda-x_1)(\lambda-p_1)}{\mathrm{diam}(E)^{n+s+2}}dx\,,
\end{equation}
and
\begin{equation}\label{e claim}
\lim_{m\rightarrow\infty}\varepsilon(m)= 0\,.
\end{equation}
First we define a new approximation of $\mathbf{H}_E^{s}(q)$.
For any $\varepsilon>0$, let
\[
f_{\varepsilon}(z):=\left\{
\begin{array}{ll}
      \varepsilon^{n+s} & \mbox{if }z\in B_\varepsilon(0)\\
      |z|^{n+s} & z\in B_\varepsilon(0)^c\,,\\
\end{array}
\right.
\]
\[
\mathbf{H}_E^{s,\varepsilon}(q):=\int_{\mathbb R^{n}}\frac{\tilde{\chi}_{E}(x)}{f_\varepsilon(|x-q|)}dx\,,
\]
and
\[
Q_m^\varepsilon:=\mathbf{H}_{\rho(E)}^{s,\varepsilon}(q_m^*)-\mathbf{H}_E^{s,\varepsilon}(q_m)\,.
\]
By the continuity of $\mathbf{H}_E^s$ we know that
\[
\lim_{\varepsilon\rightarrow 0}\mathbf{H}_E^{s,\varepsilon}(q)=\mathbf{H}_E^s(q) \qquad\forall q\in M\cap H.
\]
We have
\begin{align}\label{Q e split}
Q_m^\varepsilon=&\int_{\mathbb R^n}\frac{\tilde{\chi}_{\rho(E)}(x)}{f_{\varepsilon}(|x-q_m^*|)}dx-\int_{\mathbb R^n}\frac{\tilde{\chi}_{E}(x)}{f_{\varepsilon}(|x-q_m|)}dx\nonumber\\
=&\int_{\mathbb R^n}\frac{\tilde{\chi}_{\rho(E)}(x)-\tilde{\chi}_{E}(x)}{f_{\varepsilon}(|x-q_m^*|)}dx +\int_{\mathbb R^n}\tilde{\chi}_{E}(x)\Big(\frac{1}{f_{\varepsilon}(|x-q_m^*|)}-\frac{1}{f_{\varepsilon}(|x-q_m|)}\Big)dx\,.
\end{align}
We handle the convergence of the two integrals separately. First let
\begin{equation}\label{I R def}
I^\varepsilon_m:=\int_{\mathbb R^n}\frac{\tilde{\chi}_{\rho(E)}(x)-\tilde{\chi}_{E}(x)}{f_{\varepsilon}(|x-q_m^*|)}dx\qquad R^{\varepsilon}_m:=\int_{\mathbb R^n}\tilde{\chi}_{E}(x)\Big(\frac{1}{f_{\varepsilon}(|x-q_m^*|)}-\frac{1}{f_{\varepsilon}(|x-q_m|)}\Big)dx\,.
\end{equation}
For any $\varepsilon>0$, we claim that,
\begin{equation}\label{I claim}
\liminf_{m\rightarrow\infty} I^\varepsilon_m\ge \int_{(E\setminus \rho(E))\cap B_\varepsilon(p)^c}\frac{2(n+s)(\lambda-x_1)(\lambda-p_1)}{\mathrm{diam}(E)^{n+s+2}}dx\,.
\end{equation}
By symmetry across $\pi_\lambda$ we have
\begin{align}\label{case three to one}
I_\varepsilon=&2\int_{E\setminus \rho(E)}\frac{dx}{f_{\varepsilon}(|x-q_m^*|)}-2\int_{\rho(E)\setminus E}\frac{dx}{f_{\varepsilon}(|x-q_m^*|)}\,.\nonumber\\
=&2\int_{E\setminus \rho(E)}\Big(\frac{1}{f_{\varepsilon}(|x-q_m^*|)}-\frac{1}{f_{\varepsilon}(|\rho(x)-q_m^*|)} \Big)dx\nonumber\\
=&2\int_{(E\setminus \rho(E))\cap B_\varepsilon(q_m^*)^c}\Big(\frac{1}{f_{\varepsilon}(|x-q_m^*|)}-\frac{1}{f_{\varepsilon}(|\rho(x)-q_m^*|)} \Big)dx\nonumber\\
&+2\int_{(E\setminus \rho(E))\cap B_\varepsilon(q_m^*)}\Big(\frac{1}{f_{\varepsilon}(|x-q_m^*|)}-\frac{1}{f_{\varepsilon}(|\rho(x)-q_m^*|)} \Big)dx\,.
\end{align}
We consider each of the integrals in \eqref{case three to one} seperately. For the first integral we factor the integrand to get
\begin{equation}\label{factor I}
\frac{1}{f_{\varepsilon}(|x-q_m^*|)}-\frac{1}{f_{\varepsilon}(|\rho(x)-q_m^*|)} =\frac{1}{f_{\varepsilon}(|\rho(x)-q_m^*|)}\Big(\frac{f_{\varepsilon}(|\rho(x)-q_m^*|)}{f_{\varepsilon}(|x-q_m^*|)}-1 \Big)\,.
\end{equation}
We take $\varepsilon<(\lambda-p_1)/4$ so that $\rho(x)\in B_\varepsilon(q_m^*)^c$ for any $x\in E\setminus\rho(E)$. Then, following calculations in \eqref{case one calc two}, \eqref{case 1 mid}, and \eqref{case one pos} in case one, we know that if $x\in(E\setminus \rho(E)) \cap B_\varepsilon(q_m^*)^c $, then
\begin{equation}\label{^}
\frac{1}{f_{\varepsilon}(|\rho(x)-q_m^*|)}\Big(\frac{f_{\varepsilon}(|\rho(x)-q_m^*|)}{f_{\varepsilon}(|x-q_m^*|)}-1 \Big)\ge \frac{2(n+s)(\lambda-x_1)(\lambda-q_m^*\cdot e_1)}{\mathrm{diam}(E)^{n+s+2}}\,.
\end{equation}
Moreover if we let $\varepsilon<(\lambda-p_1)/4$ be fixed then as $m\rightarrow \infty$
\[
\chi_{B_\varepsilon(q_m^*)^c}\frac{2(n+s)(\lambda-x_1)(\lambda-q_m^*\cdot e_1)}{\mathrm{diam}(E)^{n+s+2}}\rightarrow \chi_{B_\varepsilon(p)^c}\frac{2(n+s)(\lambda-x_1)(\lambda-p_1)}{\mathrm{diam}(E)^{n+s+2}}\,.
\]
We know $(\lambda-x_1),(\lambda-q_m^*\cdot e_1)\le \mathrm{diam}(E)$, so
\[
\Big|\chi_{B_\varepsilon(q_m^*)^c}\frac{2(n+s)(\lambda-x_1)(\lambda-q_m^*\cdot e_1)}{\mathrm{diam}(E)^{n+s+2}}\Big|\le \frac{2(n+s)}{\mathrm{diam}(E)^{n+s}}\,,
\]
for all $x\in E\setminus \rho(E),$ and therefore the dominated convergence theorem implies that
\begin{equation*}
\int_{(E\setminus \rho(E))\cap B_\varepsilon(q_m^*)^c}\frac{2(n+s)(\lambda-x_1)(\lambda-q_m^*\cdot e_1)}{\mathrm{diam}(E)^{n+s+2}}dx
\end{equation*}
\[\rightarrow\int_{(E\setminus \rho(E))\cap B_\varepsilon(p)^c}\frac{2(n+s)(\lambda-x_1)(\lambda-p_1)}{\mathrm{diam}(E)^{n+s+2}}dx\,,
\]
as $m\rightarrow\infty$. Hence for the first integral in \eqref{case three to one} we have
\[
\liminf_{m\rightarrow\infty}\int_{(E\setminus \rho(E))\cap B_\varepsilon(q_m^*)^c}\Big(\frac{1}{f_{\varepsilon}(|x-q_m^*|)}-\frac{1}{f_{\varepsilon}(|\rho(x)-q_m^*|)} \Big)dx\,.
\]
\begin{equation}\label{first integrand}
\ge\int_{(E\setminus \rho(E))\cap B_\varepsilon(p)^c}\frac{2(n+s)(\lambda-x_1)(\lambda-p_1)}{\mathrm{diam}(E)^{n+s+2}}dx\,.
\end{equation}
For the second integral in \eqref{case three to one}, having $\frac{1}{f_{\varepsilon}(|\rho(x)-q_m^*|)}< \varepsilon^{-(n+s)}$, for $x\in B_\varepsilon(q_m^*)$, implies that
\[
\chi_{(E\setminus \rho(E))\cap B_\varepsilon(q_m^*)}\Big(\frac{1}{f_{\varepsilon}(|x-q_m^*|)}-\frac{1}{f_{\varepsilon}(|\rho(x)-q_m^*|)}\Big)
\]
is bounded above by $\varepsilon^{-(n+s)}$. So for fixed $\varepsilon$, the dominated convergence theorem implies that
\[
\int_{(E\setminus \rho(E))\cap B_\varepsilon(q_m^*)}\Big(\frac{1}{f_{\varepsilon}(|x-q_m^*|)}-\frac{1}{f_{\varepsilon}(|\rho(x)-q_m^*|)} \Big)dx
\]
\[
\rightarrow \int_{(E\setminus \rho(E))\cap B_\varepsilon(p)}\Big(\frac{1}{f_{\varepsilon}(|x-p|)}-\frac{1}{f_{\varepsilon}(|\rho(x)-p|)} \Big)dx\,,
\]
as $m\rightarrow \infty.$ But $f_{\varepsilon}(|\rho(x)-q_m^*|)=|\rho(x)-q_m^*|^{n+s}>2\varepsilon^{n+s}$. So for fixed $\varepsilon$ and $m$, and for $x\in (E\setminus \rho(E))\cap B_\varepsilon(q_m^*) $ we have
\begin{equation*}\begin{split}
\frac{1}{f_{\varepsilon}(|x-q_m^*|)}-\frac{1}{f_{\varepsilon}(|\rho(x)-q_m^*|)}
&=\frac{1}{\varepsilon^{n+s}}-\frac{1}{|\rho(x)-q_m^*|^{n+s}}\\
&\ge\frac{1}{2\varepsilon^{n+s}}dx\ge 0\,.
\end{split}\end{equation*}
We conclude that, for large $m,$ the second integrand in \eqref{case three to one} is positive, therefore
\begin{equation}\label{second integral}
\liminf_{m\rightarrow\infty} \int_{(E\setminus \rho(E))\cap B_\varepsilon(q_m^*)}\Big(\frac{1}{f_{\varepsilon}(|x-q_m^*|)}-\frac{1}{f_{\varepsilon}(|\rho(x)-q_m^*|)}\Big)\ge 0\,,
\end{equation}
and, by \eqref{first integrand}, we have \eqref{I claim}. We now handle the convergence of $R^\varepsilon_m$ from \eqref{I R def}. We claim that there exists $\varepsilon(m)$ such that
\begin{equation*}
\lim_{m\rightarrow\infty}\varepsilon(m)= 0
\end{equation*}
and
\begin{equation}\label{R claim}
\lim_{m\rightarrow \infty} R^{\varepsilon(m)}_m=0\,.
\end{equation}
For $\varepsilon(m)$ to be chosen, let $X_{\varepsilon(m)}=B_{\varepsilon(m)}(q_m^*)\cap B_{\varepsilon(m)}(q_m)$. Note that
\[
\int_{X_{\varepsilon(m)}}\tilde{\chi}_{E}(x)\Big(\frac{1}{f_{\varepsilon(m)}(|x-q_m^*|)}-\frac{1}{f_{\varepsilon(m)}(|x-q_m|)}\Big)dx=0
\]
because $f_{\varepsilon(m)}(|x-q_m|)=f_{\varepsilon(m)}(|x-q_m^*|)={\varepsilon(m)}^{n+s}$ in $X_{\varepsilon(m)}$.
So
\begin{equation}\label{R def simple}
R^{\varepsilon(m)}_m=\int_{X_{\varepsilon(m)}^c}\tilde{\chi}_{E}(x)\Big(\frac{1}{f_{\varepsilon(m)}(|x-q_m^*|)}-\frac{1}{f_{\varepsilon(m)}(|x-q_m|)}\Big)dx\,.
\end{equation}
However we can rewrite
\[
\int_{X_{\varepsilon(m)}^c}\frac{\tilde{\chi}_{E}(x)\;dx}{f_{\varepsilon(m)}(|x-q_m|)}=\int_{B_{\varepsilon(m)}(q_m)^c}\frac{\tilde{\chi}_{E}(x)\;dx}{f_{\varepsilon(m)}(|x-q_m|)}+\int_{B_{\varepsilon(m)}(q_m)\setminus B_{\varepsilon(m)}(q_m^*)}\frac{\tilde{\chi}_{E}(x)\;dx}{f_{\varepsilon(m)}(|x-q_m|)}\,,
\]
and similarly
\[
\int_{X_{\varepsilon(m)}^c}\frac{\tilde{\chi}_{E}(x)\;dx}{f_{\varepsilon(m)}(|x-q_m^*|)}=\int_{B_{\varepsilon(m)}(q_m^*)^c}\frac{\tilde{\chi}_{E}(x)\;dx}{f_{\varepsilon(m)}(|x-q_m^*|)}+\int_{B_{\varepsilon(m)}(q_m^*)\setminus B_{\varepsilon(m)}(q_m)}\frac{\tilde{\chi}_{E}(x)\;dx}{f_{\varepsilon(m)}(|x-q_m^*|)}\,,
\]
so, by \eqref{R def simple},
\begin{equation}\label{R bound}
\begin{split}
R^{\varepsilon(m)}_m\le &\Big|\int_{B_{\varepsilon(m)}(q_m)^c}\frac{\tilde{\chi}_{E}(x)\;dx}{f_{\varepsilon(m)}(|x-q_m|)}-\int_{B_{\varepsilon(m)}(q_m^*)^c}\frac{\tilde{\chi}_{E}(x)\;dx}{f_{\varepsilon(m)}(|x-q_m^*|)}\Big|\\
&+\Big|\int_{B_{\varepsilon(m)}(q_m)\setminus B_{\varepsilon(m)}(q_m^*)}\frac{\tilde{\chi}_{E}(x)\;dx}{f_{\varepsilon(m)}(|x-q_m|)}\Big|+\Big|\int_{B_{\varepsilon(m)}(q_m^*)\setminus B_{\varepsilon(m)}(q_m)}\frac{\tilde{\chi}_{E}(x)\;dx}{f_{\varepsilon(m)}(|x-q_m^*|)}\Big|\,.
\end{split}
\end{equation}
Note that there is a universal constant $C(n)$, dependent only on $n$, such that, for all $r>0$, we know
\[
|B_r(w)\Delta B_r(0)|\le C(n)r^{n-1}|w| \qquad \forall w\in\mathbb R^n\,.
\]
Therefore, we have
\begin{equation}\label{3 main 1}
\int_{B_{\varepsilon(m)}(q_m)\setminus B_{\varepsilon(m)}(q_m^*)}\frac{\tilde{\chi}_{E}(x)\;dx}{f_{\varepsilon(m)}(|x-q_m|)}\le\frac{|B_{\varepsilon(m)}(q_m^*)\Delta B_{\varepsilon(m)}(q_m)|}{2{\varepsilon(m)}^{n+s}}\le\frac{C(n)|q_m-q_m^*|}{{\varepsilon(m)}^{s+1}}\,,
\end{equation}
and
\begin{equation}\label{3 main 2}
\int_{B_{\varepsilon(m)}(q_m^*)\setminus B_{\varepsilon(m)}(q_m)}\frac{\tilde{\chi}_{E}(x)\; dx}{f_{\varepsilon(m)}(|x-q_m^*|)}\le \frac{|B_{\varepsilon(m)}(q_m^*)\Delta B_\varepsilon(q_m)|}{2{\varepsilon(m)}^{n+s}}\le\frac{C(n)|q_m-q_m^*|}{{\varepsilon(m)}^{s+1}}\,.
\end{equation}
Also, by applying \cite[Remark 3.25]{AmbFusPal} with $u=\chi_E$,
\[
\Big|\int_{B_{\varepsilon(m)}(q_m)^c}\frac{\tilde{\chi}_{E}(x)}{f_{\varepsilon(m)}(|x-q_m|)}dx-\int_{B_{\varepsilon(m)}(q_m^*)^c}\frac{\tilde{\chi}_{E}(x)}{f_{\varepsilon(m)}(|x-q_m^*|)}dx\Big|\quad
\]
\begin{align}\label{3 main 3}
&=\Big|\int_{B_{\varepsilon(m)}(0)^c}\Big(\frac{\tilde{\chi}_{E}(x+q_m)-\tilde{\chi}_{E}(x+q_m^*)}{|x|^{n+s}}\Big)dx\Big|\nonumber\\
&\le\int_{B_{\varepsilon(m)}(0)^c}\frac{\chi_{(E+q_m)\Delta(E+q_m^*)}}{|x|^{n+s}}dx\nonumber\\
&\le \frac{P(E)|q_m-q_m^*|}{{\varepsilon(m)}^{n+s}}\,.
\end{align}
We choose $\b>0$, and set $\varepsilon(m)=|q_m-q_m^*|^{1/(n+s+\b)}$. Note that \eqref{e claim} holds because
\[
|q_m-q_m^*|=O\big((1/m)^{2+\a})\,.
\]
Moreover, for this choice of $\varepsilon(m)$, we also have that \eqref{3 main 1}, \eqref{3 main 2}, and \eqref{3 main 3} converge to $0$ as $m\rightarrow\infty,$ and therefore with \eqref{R bound}, we have \eqref{R claim}. By \eqref{e claim}, by \eqref{I claim}, and because of our choice of $\varepsilon(m)$, we have
\begin{equation}\label{I claim real}
\liminf_{m\rightarrow\infty} I^{\varepsilon(m)}_m\ge \int_{E\setminus \rho(E)}\frac{2(n+s)(\lambda-x_1)(\lambda-p_1)}{\mathrm{diam}(E)^{n+s+2}}dx\,.
\end{equation}
Hence, by \eqref{Q e split}, together \eqref{I claim real} and \eqref{R claim} imply \eqref{Step Two}.

\smallskip

\noindent {\it Step three}:
We show that
\begin{equation}\label{switch order}
\liminf\limits_{m\rightarrow \infty}Q_m^{\varepsilon(m)}=\liminf\limits_{m\rightarrow \infty}\mathbf{H}_{\rho(E)}^{s,\varepsilon(m)}(q_m^*)-\mathbf{H}_E^{s,\varepsilon(m)}(q_m)=\lim\limits_{m\rightarrow \infty} \mathbf{H}_{\rho(E)}^s(q_m^*)-\mathbf{H}_E^{s}(q_m)=\lim\limits_{m\rightarrow \infty}Q_m\,,
\end{equation}
to get \eqref{int diff} in case three. To see this convergence we show that
\begin{equation}\label{conv 331}
\lim\limits_{m\rightarrow \infty} |\mathbf{H}_E^{s,\varepsilon(m)}(q_m)-\mathbf{H}_E^s(q_m)|=0\,,
\end{equation}
and equivalently that
\begin{equation}\label{conv 332}
\lim\limits_{m\rightarrow \infty} |\mathbf{H}_{\rho(E)}^{s,\varepsilon(m)}(q_m^*)-\mathbf{H}_{\rho(E)}^s(q_m^*)|=0\,.
\end{equation}
Here we use the properties of the graphs locally defining $M$ and $\rho(M)$. By definition,
\begin{equation*}
\big|\mathbf{H}_{E}^{s,\varepsilon(m)}(q_m)-\mathbf{H}_E^s(q_m)\big|=\Big|\int_{B_{\varepsilon(m)}(q_m)}\tilde{\chi}_E(x)\Big[\frac{1}{|x-q_m|^{n+s}}-\frac{1}{\varepsilon(m)^{n+s}}\Big]dx\Big|\,.
\end{equation*}
Because $\varepsilon(m)\rightarrow 0$ as $m\rightarrow\infty$, we know that $\varepsilon(m)<\eta$, for $m$ large enough and for $\eta$ as in the definition of $P_{\eta,\gamma}(q)$ in \eqref{parabola}. So, for $m$ large enough, we have
\begin{equation*}\begin{split}
\big|\mathbf{H}_E^{s,\varepsilon(m)}(q_m)-\mathbf{H}_E^s(q_m)\big|&=\Big|\int_{ P_{\varepsilon(m),\gamma}(q_m)}\tilde{\chi}_E(x)\Big(\frac{1}{|x-q_m|^{n+s}}-\frac{1}{\varepsilon(m)^{n+s}}\Big)dx\Big|\\
&\le\int_{ P_{\varepsilon(m),\gamma}(q_m)}\frac{1}{|x-q_m|^{n+s}}dx\,,
\end{split}\end{equation*}
which goes to 0 as $m\rightarrow \infty.$ Therefore we have \eqref{conv 331} and \eqref{conv 332}, and thus \eqref{switch order} as claimed.
Hence, by \eqref{Step Two} from step two we have
\[
\lim\limits_{m\rightarrow \infty} Q_m\ge\int_{E\setminus \rho(E)}\frac{2(n+s)(\lambda-x_1)(\lambda-p_1)}{\mathrm{diam}(E)^{n+s+2}}dx\,,
\]
which combined with \eqref{*} from step one, implies
\[
\frac{5|\lambda-p_1|\delta_s(E)}{2\mathrm{diam}(E)^{s+1}}\ge\int_{E\setminus \rho(E)}\frac{2(n+s)(\lambda-x_1)(\lambda-p_1)}{\mathrm{diam}(E)^{n+s+2}}dx\,.
\]
Therefore, since $\lambda-p_1>0,$ we have \eqref{int diff} in case three.

\medskip

\noindent\textit{Case four}: In this case we assume that $p\in \pi_\lambda\cap{\rm bd}\,(M).$ As in case three we use the local graphicality of $M$ and $\rho(M)$ near $p$ to find a sequence of points to approximate the work done in case two. We want a sequence of points approaching $p$ in $M\cap \pi_\lambda=\rho(M)\cap \pi_\lambda.$ Let $\hat{q}_m:=p-\nu_{co}^M(p)/m,$ where $m\in \mathbb N$ and  $\nu_{co}^M(p)$ is the conormal vector for $p$ with respect to $M.$ Set $q_m=v(\hat{q}_m)$ for $v$ as in case three. Let $u_{\varepsilon,q}(x):=\varphi_\varepsilon(|x-q|)$ and let
\[
\mathbf{H}_E^{s,\varepsilon}(q):={\rm p.v.}\int_{\mathbb R^n}\tilde{\chi}_{E}(x)\varphi_\varepsilon(|x-q|)dx\,,
\]
as in case two. For any $\t_m\in T_{q_m}M$ such that $\t_m\cdot e_n=0$, by \cite[Lemma 2.1]{ciraolofigallimagginovaga}, we know
\[
\nabla \mathbf{H}_E^{s,\varepsilon}(q_m)\cdot \t_m= \int_{\mathbb R^n}\tilde{\chi}_E(x)\nabla u_{\varepsilon,q_m}(x)\cdot \t_m\;dx\,.
\]
Let $P_m$ be the hyperplane through $q_m$ which is perpendicular to $\t_m.$ Set $E_m^*$ to be the reflection of $E$ across $P_m.$ Then, because $\nabla u_{\varepsilon,q_m}(x)$ is odd with respect to $P_m,$ and because $\t_m\cdot e_n=0$, we have
\[
\nabla \mathbf{H}_E^{s,\varepsilon}(q_m)\cdot \t_m=- 2\int_{E\setminus E_m^*}\nabla u_{\varepsilon,q_m}(x)\cdot \t_m \;dx\,.
\]
But because $\nabla u_{\varepsilon,q_m}(x)\cdot \t_m=\varphi'_\varepsilon(|x-q_m|)\frac{(x-q_m)\cdot \t_m}{|x-q_m|}$ we know
\begin{equation*}\begin{split}
\nabla \mathbf{H}_E^{s,\varepsilon}(q_m)\cdot \t_m&=- 2\int_{E\setminus E_m^*}\varphi'_\varepsilon(|x-q_m|)\frac{(x-q_m)\cdot \t_m}{|x-q_m|}dx\\
&=- 2\int_{\mathbb R_n}\chi_{E\setminus E_m^*}(x)\varphi'_\varepsilon(|x-q_m|)\frac{(x-q_m)\cdot \t_m}{|x-q_m|}dx\,.
\end{split}\end{equation*}
Therefore, by the monotone convergence of $|\varphi'_\varepsilon|$ in $\varepsilon$
\begin{equation}\label{tangent limit}
\lim_{\varepsilon\rightarrow 0}\nabla\mathbf{H}_E^{s,\varepsilon}(q_m)\cdot \t_m=-2(n+s)\int_{\mathbb R^n}\chi_{E\setminus E^*_m}(x)\frac{(q_m-x)\cdot \t_m}{|x-q_m|^{n+s+2}}dx\,.
\end{equation}
Next, set
\[
\t_m:=e_1-(w_m\cdot e_1)\frac{w_m}{||w_m||}
\]
with
\[
w_m:=\nu_{E}(q_m)-(\nu_{E}(q_m)\cdot e_n)e_n\,.
\]
Then $\t_m\rightarrow e_1$ and $\chi_{E\setminus E_m^*}\rightarrow \chi_{E\setminus \rho(E)}$ pointwise in $m$ because $\nu_E(p)\cdot e_1=0$ and $\nu_E(q_m)\rightarrow \nu_E(p)$ as $m\rightarrow\infty$ imply that $w_m\cdot e_1\rightarrow 0$ as $m\rightarrow\infty$. So
\[
\lim\limits_{m\rightarrow\infty}\chi_{E\setminus E^*_m}(x)\frac{(q_m-x)\cdot \t_m}{|x-q_m|^{n+s+2}}=\chi_{E\setminus \rho(E)}(x)\frac{(p-x)\cdot e_1}{|x-p|^{n+s+2}}
\]
pointwise. By definition, $\t_m\cdot e_n=0$ and $\t_m\in T_{q_m}M$. Therefore, by applying Fatou's lemma to \eqref{tangent limit}, we have
\[
\liminf\limits_{m\rightarrow\infty}\lim_{\varepsilon\rightarrow 0}-\nabla\mathbf{H}_E^{s,\varepsilon}(q_m)\cdot \t_m\ge\int_{\mathbb R_n}\chi_{E\setminus \rho(E)}(x)\frac{(p-x)\cdot e_1}{|x-p|^{n+s+2}}dx=\int_{E\setminus \rho(E)}\frac{(p-x)\cdot e_1}{|x-p|^{n+s+2}}dx\,.
\]
Thus, because $\frac{(p-x)\cdot e_1}{|x-p|^{n+s+2}}$ is positive by construction and
\[
\delta_s(E)\ge\mathrm{diam}(E)^{s+1}|\nabla \mathbf{H}_E^{s,\varepsilon}(q_m)\cdot \t_m|\,,
\]
 for all $m$ large, we have \eqref{int diff} in case four.

\smallskip

\noindent\textit{Conclusion to part (a)}:
We have concluded in every case that
\begin{equation*}
\int_{E\Delta \rho(E)}\hbox{dist}(x,\pi_\lambda)dx\le \frac{5}{2(n+s)}\mathrm{diam}(E)^{n+1}\delta_s(E)\,.
\end{equation*}
Thus, by Chebyshev's inequality, for any $\b>0$,
\[
\big|\{x\in E\Delta\rho(E):\hbox{dist}(x,\pi_\lambda)\ge \b\}\big|\le\frac{1}{\b}\frac{5}{2(n+s)}\mathrm{diam}(E)^{n+1}\delta_s(E)\,.
\]
So because
\[
\big|\{x\in E\Delta\rho(E):\hbox{dist}(x,\pi_\lambda)\le \b\}\big|\le 2\b\mathrm{diam}(E)^{n-1}\,,
\]
by setting
\[
\b=\sqrt{\frac{5}{2(n+s)}}\mathrm{diam}(E)\sqrt{\delta_s(E)}\,,
\]
we have
\begin{equation*}
|E\Delta\rho (E)|\le C_1\mathrm{diam}(E)^{n}\sqrt{\delta_s(E)}\,,
\end{equation*}
for
\begin{equation}\label{constant one}
C_1=3\sqrt{\frac{5}{2(n+s)}}\,.
\end{equation}
\end{proof}

\section{Improved Understanding of the Geometry}
We will use the following lemma to prove Theorem \ref{thm axially symmetric}\ref{improved thm}. The proof closely follows the proof of \cite[Lemma 4.1]{ciraolofigallimagginovaga}.

\begin{lemma}\label{l bound 1} Let $E$ be as in assumption (h1) with
\begin{equation}\label{delta bound}
\sqrt{\delta_s(E)}\le \frac{1}{3} \min\Big\{\frac{1}{2},\frac{1}{n-1}\Big\}\sqrt{\frac{2(n+s)}{5}}\frac{|E|}{\mathrm{diam}(E)^{n}}\,.
\end{equation}
Suppose that the critical planes with respect to the coordinate directions, coincide with $\{x_i=0\}$ for all $i=1,...,n-1$. If $e\in S^{n-1}\cap\partial H$, and $\lambda_e$ is the critical value associated with $e$ in \eqref{lambda}, then
\begin{equation*}\label{lambda bound}
|\lambda_e| \le C\frac{\mathrm{diam}(E)^{n+1}}{|E|}\sqrt{\delta_s(E)}
\end{equation*}
holds for some constant $C=C(n).$
\end{lemma}
\begin{proof}
We begin by letting $F^0:=\{(-\hat{x},x_n):x\in F\}$ for $F\subset \mathbb R^n$. Set
\[
C_1^*:=C_1\mathrm{diam}(E)^{n}\,.
\]
with $C_1$ as defined in \eqref{constant one} from the proof of Theorem \ref{thm axially symmetric}(a). Theorem~\ref{thm axially symmetric}(a) implies
\begin{equation}\label{combined thm}
|E\Delta E^0|\le (n-1)C_1^*\sqrt{\delta_s(E)}\,,
\end{equation}
because $E^0$ can be obtained from $E$ by symmetrizing with respect to the $n-1$ hyperplanes $\{x_i=0\}$ for $i=1,..,n-1$.
So by \eqref{delta bound} we have $|E\Delta E^0|\le |E|.$ Without loss of generality we assume $\lambda_e>0$. Note that
\begin{equation}\label{big L bound}
\L_e:=\sup\{x\cdot e:x\in E\}\le \mathrm{diam}(E)\,.
\end{equation}
If not, we would have $x\cdot e\ge 0$ for all $x\in E,$ which would imply that $|E\Delta E^0|=2|E|$. Moreover, by Theorem~\ref{thm axially symmetric}(a) we also have
\begin{equation}\label{refl bound}
|E\Delta \rho(E)|\le C_1^*\sqrt{\delta_e(E)}\,,
\end{equation}
with $\rho$ defined to be the function that reflects $E$ across the critical hyperplane $\pi_e$.
So
\begin{equation}\label{most mass}
|E\cap\{x\cdot e>\lambda_e\}|\ge\frac{1}{2}(|E|-C_1^*\sqrt{\delta_s(E)})\,.
\end{equation}
Then, by \eqref{combined thm} and \eqref{most mass}, we have
\[
|E\cap \{x\cdot e>\lambda_e\}^0|=|E^0\cap \{x\cdot e>\lambda_e\}|\ge |E\cap\{x\cdot e>\lambda_e\}|-|E\Delta E^0|\ge\frac{|E|}{2}-\Big(n-\frac{1}2\Big)C_1^*\sqrt{\delta_s(E)}\,,
\]
so, by applying \eqref{most mass} again, we know
\begin{align}\label{small strip}
|\{x\in E:-\lambda_e\le x\cdot e = \lambda_e\}|\le&\, |\{x\in E:x\cdot e<0\}|-|E\cap \{x\cdot e>\lambda_e\}^0|\nonumber \\
&+ |\{x\in E:x\cdot e>0\}| -|E\cap\{x\cdot e>\lambda_e\}|  \nonumber\\
\le&\, nC_1^*\sqrt{\delta_s(E)}\,.
\end{align}
We have shown that $E$ has small volume in the strip $\{|x\cdot e|<\lambda_e\}.$ We continue by quantifying the volume of $E$ in parallel strips of the same width. The set $\{\lambda_e\le x\cdot e\le 3\lambda_e\}$ is mapped into $\{|x\cdot e|<\lambda_e\}$ by reflection with respect to the critical hyperplane, so by \eqref{refl bound} and \eqref{small strip} we have
\begin{equation*}\begin{split}
|\{x\in E:\lambda_e\le x\cdot e \le 3\lambda_e\}|&=|\{x\in \rho(E):|x\cdot e| \le \lambda_e\}|\\
&\le|\{x\in E:|x\cdot e| \le \lambda_e\}|+|E\Delta \rho(E)|\le (n+1)C_1^*\sqrt{\delta_s(E)}\,.\\
\end{split}\end{equation*}
Now let
\[
m_k:=|\{x\in E:(2k-1)\lambda_e\le x\cdot e\le (2k+1)\lambda_e\}|, \qquad k\ge 1\,.
\]
By the moving planes procedure, if $\lambda_e\le \mu'\le\mu$, then $E\cap \pi_{\mu}\subset E\cap\pi_{\mu'}$, where each set is seen as a subset in $\mathbb R^{n-1}.$ So because $\mathcal H^{n-1}(E\cap\pi_\mu)$ is decreasing in $\mu$, for $\mu\in(\lambda_e,\L_e)$, it follows that $m_k$ is decreasing in $k.$ Therefore
\begin{equation}\label{m gets smaller}
m_k\le m_1\le (n+1)C_1^*\sqrt{\delta_s(E)}\qquad k\ge 1\,.
\end{equation}
Let $k_0$ be the smallest natural number such that $(2k_0+1)\lambda_e\ge \L_e,$ which implies $(2k_0-1)\lambda_e\le \L_e.$  Because  $E\subset\{x\cdot e\le \L_e\}$, by \eqref{m gets smaller} we see that
\begin{equation*}
|E\cap\{x\cdot e>\lambda_e\}|=|E\cap\{\lambda_e\le x\cdot e\le \L_e\}|=\sum_{k=1}^{k_0}m_k\le \frac{1}2\Big(\frac{\L_e}{\lambda_e}+1\Big)(n+1)C_1^*\sqrt{\delta_s(E)}\,.
\end{equation*}
Then \eqref{big L bound} implies
\begin{equation}\label{almost}
|E\cap\{x\cdot e>\lambda_e\}|\lambda_e\le (n+1)C_1^*\mathrm{diam}(E)\sqrt{\delta_s(E)}\,.
\end{equation}
Lastly, by (\ref{delta bound}) and (\ref{most mass}), we know $|E\cap\{x\cdot e>\lambda_e\}|\ge|E|/4$, which combined with  \eqref{almost} and the definition of $C_1$ implies
\begin{equation}\label{real lambda bound}
\lambda_e\le \frac{(n+1)C_1^*\mathrm{diam}(E)\sqrt{\delta_s(E)}}{|E\cap\{x\cdot e>\lambda_e\}|}\le 4(n+1)C_1\frac{\mathrm{diam}(E)^{n+1}}{|E|}\sqrt{\delta_s(E)}\,,
\end{equation}
as required.\end{proof}

\begin{proof}[Proof of Theorem 1.1 (b)] Let
\[
\delta_0=\frac{1}{3} \min\Big\{\frac{1}{2},\frac{1}{n-1}\Big\}\sqrt{\frac{2(n+s)}{5}}\frac{|E|}{\mathrm{diam}(E)^{n}}.
\] We assume that $\delta(E)\le \delta_0$, so Lemma \ref{l bound 1} applies. Up to translation, we may also assume that the critical planes with respect to the coordinate directions $e_i$ coincide with $\{x_i=0\}$ for every $i=1,...,n-1.$
Let $E_h$ be the cross section of $E$ parallel to $\partial H$ at height $h.$ For each height $h$ let
\[
r_h=\inf\limits_{x\in \partial E_h}|x-he_n|\qquad\mbox{ and }\qquad R_h=\sup\limits_{x\in \partial E_h}|x-he_n|\,.
\]
Choose $x_h,y_h$ such that $|x_h-h e_n|=r_h$ and $|y_h-he_n|=R_h$. Without loss of generality we may assume that $x_h\ne y_h$, otherwise $R_h-r_h=0$. Let
\[
e_h:=\frac{y_h-x_h}{|y_h-x_h|}\,,
\]
let $\lambda_h=\lambda_{e_h}$ be the critical value for $e_h,$ and let $\pi_{h}=\pi_{\lambda_{e_h}}$ denote the critical hyperplane.
 Note that $y_h$ is closer to $\pi_h$ than $x_h$ in $\{x_n=h\}$, that is,
\begin{equation}\label{dist plane}
\hbox{dist}(x_h,\pi_{h})\ge\hbox{dist}(y_h,\pi_{h})\,.
\end{equation}
Indeed, by the moving planes method, the critical position can be reached at most when $\rho(y_h)$, the reflection of $y_h$ with respect to $\pi_h$, is $x_h$. In this case we have equality in \eqref{dist plane}, but otherwise we have strict inequality. So by \eqref{dist plane}, and because $e_h$ is parallel to $y_h-x_h,$ we have
\begin{equation}\label{rs diff}\begin{split}
R_h-r_h&=|y_h-he_n|-|x_h-he_n|\le |(y_h-h e_n)-(x_h-h e_n)|\\
&=(\hbox{dist}(y,\pi_{h})+\lambda_{h})-(\hbox{dist}(x,\pi_{h})-\lambda_{h})\\
&\le2|\lambda_{h}|\,.
\end{split}\end{equation}
Set $C_2=4(n+1)C_1$, with $C_1$ from \eqref{constant one}. Combining \eqref{real lambda bound} with \eqref{rs diff}, we get
\begin{equation*}\label{radii dist}
R_h-r_h\le 2C_2\frac{\mathrm{diam}(E)^{n+1}}{|E|}\sqrt{\delta_s(E)}\,,
\end{equation*}
or equivalently
\begin{equation*}\label{radii dist}
\frac{R_h-r_h}{\mathrm{diam}(E)}\le 2C_2\frac{\mathrm{diam}(E)^{n}}{|E|}\sqrt{\delta_s(E)}\,,
\end{equation*}
which is \eqref{R diff bound}.

Note that  \eqref{R diff bound} just implies that the boundary of $E_h$ is contained in an annulus with radii $r_h$ and $R_h$. In fact, if $\mathrm{diam}(E_h)$ is small enough then $E_h$ could be contained in the annulus as well, and may not contain the inner ball $D_{h}:=B_{r_h}(he_n)\cap\{x_n=h\}$. However, we will show that if
\begin{equation}\label{slice di bound}
\frac{\mathrm{diam}(E_h)}{\mathrm{diam}(E)}>  6C_2\frac{\mathrm{diam}(E)^{n}}{|E|}\sqrt{\delta_s(E)}\,,
\end{equation}
then $D_h\subset E_h$. Suppose that $D_{h}$ is not contained in $E_h$.
By applying the moving planes argument in any direction $e$ such that $e\cdot e_n=0$ and $\{x\in E_h:e\cdot x/|x|=\pm1\}\ne\emptyset$, we know that
\[
|\lambda_e|\ge r_h.
\]
Together with \eqref{real lambda bound} this implies
\begin{equation*}\label{small di r bound}
r_h\le C_2\frac{\mathrm{diam}(E)^{n+1}}{|E|}\sqrt{\delta_s(E)}\,,
\end{equation*}
so
\begin{equation*
\mathrm{diam}(E_h)\le 2R_h\le 2\Big(2C_2\frac{\mathrm{diam}(E)^{n+1}}{|E|}\sqrt{\delta_s(E)}+r_h\label{small enough di}\Big)\le  6C_2\frac{\mathrm{diam}(E)^{n+1}}{|E|}\sqrt{\delta_s(E)}\,,
\end{equation*}
which is equivalent to \eqref{slice di bound}.
In particular, combining this inequality with $\mathrm{diam}(E_h)\ge R_h-r_h$ gives
\[
R_h\le \mathrm{diam}(E_h)+r_h\le 7C_2\frac{\mathrm{diam}(E)^{n+1}}{|E|}\sqrt{\delta_s(E)}\,,
\]
which concludes the proof of the theorem.
\end{proof}

redpajama_set_name: RedPajamaArXiv


text:
Q: IP white list for docker containers with iptables and ipset I want to restrict access to docker container ports just from specified IPs.
I set up iptables rules with ipset.
I have exposed the port 8888. The requests from port 8888 are forwarded to simple docker web server.
I defined ipset with white list IP addresses.
ipset create testfilter iphash
ipset add testfilter 192.168.52.65

Then I created iptables rules. New chain with name testfilter, FORWARD rule with port 8888 should jump to testfilter. The first rule from testfilter chain should match IP from white list ipset. The second rule from testfilter chain should dropping communication when IP doesn't match with white list.
iptables -N testfilter
iptables -I FORWARD -p tcp --dport 8888 -j testfilter
iptables -A testfilter -m set --match-set test_ips src -j RETURN
iptables -A testfilter -j DROP

But port 8888 is still accessable from any IP. I also try to use INPUT rule instead of FORWARD rule, also remove --dport param from rule and put -m state --state NEW param instead of that. Any ideas how to fix that?

A: You can access your 8888 port because your docker container is running on localhost and the FORWARD isn't blocking it. Please use -i option to specify lo interface.

redpajama_set_name: RedPajamaStackExchange


John (Jack) Christy was born 20 August 1911 in Ashwell the son of Harry Oakley Christy and Louisa Patey who owned and ran Christy's (Ashwell Stores). For more information about the Christy family and shop please refer to the 'Places' 'Shops' headings on this website.
Jack attended The Grammar School, Hitchin in the early 1920s.
Jack married Myra Anderson in 1933 later divorcing.
He served as a flight Sgt in the RAF during WWII in the Pigeon Service spending time at the Sandringham Royal Lofts and later in Iran and Egypt.
Jack married Florence Grace Boddy in 1947. Florence served in the WRAF during WWII as a Sgt Barrage Balloon Operator.
This page was added on 29/10/2013.


The concept of bisexuality has certainly had an adequate amount of attention from scholars and researchers around the globe. Different theories and models have been used to explain how it occurs in the life of a man. In this chapter, bisexuality is viewed in terms of various theories such as psychoanalytic theory, development theory, and sexual orientation theory to help explain its shift from the transitional event it used to be viewed as to the distinct sexual orientation as we perceive it today. The theories show that individuals arrive at their sexual identities after taking several probable routes that can be stimulated by social and personal influences. In the surveys conducted a more accurate characterization of experimental groups manifests if participants are sufficiently discriminated by sexual orientation. The chapter also gives empirical data that describe patterns, similarities, and differences in the development of bisexuals, lesbians, and gay identities.


text:
January 22, 2004 adminrestoration
January 22, 2004 Happy Chinese New Year! It's the year of the Monkey, and since Cherokee's rebirth is spring of '04, she shares traits with the monkey – agile, quick, and smart. She can get in and out of tight places, and plays well with humans. Keith Rittmaster, head of the NC Maritime Museum's Cape Lookout Studies Program, snapped this shot of Cherokee off the Cape New Year's week.
Huckins Sighting off Cape Lookout
Meanwhile, back at the berth, Bryan and Chris laid the galley flooring and trim. Below left they're placing the last bit of trim, and right shows the completed floor. The empty area to the right is where the refrigerators go.
Laying Down the Groove
Galley Gleaming
The Varnish Vixens are working their way through the galley trim, sanding and sanding in preparation for many coats of varnish. Sharon and Caroline are padding about in socks, keeping everything scratch free and pretty. Below right, Carolyn's mother Margaret is in the facility sanding a galley cabinet. Speaking of the facility, it's no longer a gaping void since the Cherokee launch. Lloyd Pigott had George lift the CBCII (Core Banks Club), an old ferry boat, into the shed whereupon it'll get converted into a houseboat (see bottom photo).
Sanding, sanding, sanding
Tom Parker Fitting Stateroom Shelf
Tom Parker, above, is building a storage cabinet in the forward stateroom over the bunk. Bryan and Chris glued the teak on the first hatch of the cockpit, and attached the hinge. They and Leonard are shown below fitting it in place.
Cockpit Hatch Placed over Generator
Chris and Bryan started the cockpit stairs, designed to be used to enter the cockpit from the aft-deck or the dock. Once completed, the stairs will have teak treads and white risers. Below Bryan and son James check it out.
Behold the dogfish – neither a dog nor a true fish, but a kind of mini-shark. The spiny dogfish, or Squalus acanthias, is the most abundant shark species, and smallest in size. It only grows to about 35 inches, and tends to travel in schools segregated by sex and age. Commercial fishermen considered dogfish "trash" until the little sharks got popular in Europe for fish and chips. National Marine Fisheries Service and environmental groups became concerned about harvest increases, so restrictions took effect in 2000, cutting the take from about 40 million pounds per year to four million pounds. North Carolina fishermen, who had been the second largest harvestor of dogfish, were cut out from the fishery entirely because Massachusetts caught the entire year's quota before the dogfish appeared in NC waters. Then in 2003 the Atlantic States Marine Fisheries Commission divided the quota between the north and the south, giving Tarheel fishermen a shot at 2 million pounds. Trouble was, the dividing line was established below Massachusettes, making the "southern" region Rhode Island to Florida! And setting the quota so low increased the amount of discarded dogfish, caught as bycatch to gray trout, croaker, etc. Last year scientists calculated that in addition to the four million pounds of dogfish that were landed, a whopping 19 million pounds were tossed overboard, about half of which were dead! "Wasting fish is not conservation," said fish dealer Willie Etheridge of Wanchese. "Increase the quota so we can at least utilize our bycatch!" Fishermen report a burgeoning population of dogfish since the quota was set, making it hard to avoid the little sharks. Scientists, however, say stocks aren't projected to recover for another 23 years. Fishermen fear that the discard problem will be addressed not by allowing them to keep dead dogfish, but by shutting down all the other fisheries that result in dogfish bycatch. "Meanwhile we're losing our industry's chance to harvest these fish to Canada," said NC fish processor and fisheries commissioner Tilman Gray. "It makes no sense." The latest twist in the dogfish saga is a permit fiasco. Tarheel fishermen can land 2,000 pounds of dogfish per day in North Carolina during the winter season only by cancelling their federal permit which limits them to 300 pounds. After the season, NMFS invites them to reapply for their federal permit. "North Carolina has to step up and do something!" said Gray. Either fight for equity and non-waste of the resource, or issue straight-jackets to the fishermen, because dogfish management is driving them crazy! Signing off for now, Barbara "Fish Doctor" Blake

redpajama_set_name: RedPajamaCommonCrawl


text:
Government remains committed to defend the interests and protect the rights of Maltese and Gozitan hunters and trappers
Posted by Vento on 4th December 2020 in Local News
The Maltese Government has taken note of the two Letters of Formal Notice sent by the European Commission regarding the research derogation to allow the trapping of finches in Malta and the enforcement of the general system of protection of wild birds and of derogations to allow trapping and hunting of wild birds in Malta.
Whilst noting that replies will be provided within the stipulated timeframe, the Government remains committed to defend the interests and protect the rights of Maltese and Gozitan hunters and trappers, as these rights are protected in various other European Member States.
The Government is convinced that the derogations applied are in conformity with the European Union's Birds Directive and it is to be noted that a higher system of enforcement is applied locally beyond that demanded by the European Commission itself.
With regard to the trapping of Golden Plover and Song Thrush, the Maltese Government recalls that it was the European Commission itself that requested amendments to the national legal framework in order to conform with the decision of the Court of Justice of the European Union on the trapping of songbirds. The Government also recalls that it was the Court of Justice itself that declared, in 2009, that in the case of Malta, there is no other solution to the hunting of pigeons and quail in Malta.
In November 2019, the European Commission asked the Maltese authorities to further analyse the reference population of the two species of bird. The Ornis Committee subsequently unanimously agreed that a research project on the reference population of Golden Plover and Song Thrush takes place.
The Maltese Government remains committed to continue discussing the matter with the European Commission in order to clarify the various divergences, whilst ensuring that the traditional hobby of hunters and trappers remains protected with full respect to the legislation in place and not accepting any abuse.
Posted in Local News | Tagged derogation, European Commission, government of Malta, hunters, Malta, trappers
Scottish court upholds Libyan Lockerbie bomber's conviction→
Hundreds unite to save Dingli trees→
'Towards a New Prosperity': Preparing our businesses wisely and cautiously for the months and years to come→
HSBC sparks outrage over newly launched bank charges→


Was there any error that happen when you tried modifying the Checklist option?
Can you provide us more details. Which of the Checklist in the form is having the issue?
I tried it on your jotform http://www.jotformpro.com/form/50542645203952 but I could not reproduce any issue. I was able to add more option on the checklist.
Please see the screen below when I test it.
This happen to me also with another widget when you update it..
Thank you for providing us with more details and screenshot.
So far I could not replicate the problem on my end. But obviously it happens for you.
I will elevate this thread to our higher team support so that it can be immediately check.
I'm sorry about the inconvenience this caused you.
This happened when we changed data centers for operational reasons - and your widgets were loaded from an older version.
I have refreshed all your widgets to access the current version - so you should be able to access all settings again - please check.


Virtual tour close by Old Christchurch Lane in Bournemouth, will let you explore building, your are looking for and its neighbourhood. The panorama is interactive so you can explore HQ foto of address, you were looking for. Turn around by 360 deg., walk along the street back and forth. Use your mouse to control panorama.


text:
61580 Baby Blues 2012
Search Results for: 61580 baby blues 2012
Dr. Adrian Helmsley, part of a worldwide geophysical team investigating the effect on the earth of radiation from unprecedented solar storms, learns that the earth's core is heating up. He warns U.S. President Thomas Wilson that the crust of the earth is becoming unstable and that without proper preparations for saving a fraction of the world's population, the entire race is doomed. Meanwhile, writer Jackson Curtis stumbles on the same information. While the world's leaders race to build "arks" to escape the impending cataclysm, Curtis struggles to find a way to save his family. Meanwhile, volcanic eruptions and earthquakes of unprecedented strength wreak havoc around the world.
Action | Adventure | Science Fiction
2012: Supernova 2009
Two hundred years ago a supernova exploded somewhere in the Lyra constellation. Now the lethal burst of radiation is headed straight for Earth, and time is swiftly running out. The only thing standing between humanity and complete devastation is astrophysicist Dr. Kelvin (Brian Krause), who heads up a project to save the planet.
Action | Adventure | Science Fiction | Thriller
2012 Doomsday 2008
On December 21, 2012 four strangers on a journey of faith are drawn to an ancient temple in the heart of Mexico. For the Mayans it is the last recorded day. For NASA scientists it is a cataclysmic polar shift. For the rest of us, it is Doomsday.
Action | Adventure | Fantasy | Horror | Science Fiction | Thriller
2012: Ice Age 2011
A volcanic eruption in Iceland sends a glacier towards North America, causing everything in it's path to freeze. A family man struggles to escape the onslaught of the coming ice age.
Pusher 2012 2012
In London, a drug dealer grows increasingly desperate over the course of a week after a botched deal lands him in the merciless clutches of a ruthless crime lord. The more desperate his behavior, the more isolated he becomes until there is nothing left standing between him and the bullet his debtors intend to fire his way.
Crime | Thriller
Beautiful 2012 2012
A superb package of shorts by four leading East Asian directors: Ann Hui on a male-to-female sex change, Kim Tae-yong on an emotional imposture, Gu Changwei on pregnancy in China and Tsai Ming-Liang on time and the city of Hong Kong.
Deception and foul play have brought CM Punk to the brink of a year-long title reign. But can he survive with the odds stacked against him as he defends his coveted WWE Championship against Ryback and John Cena in a Triple Threat Match? Plus, the Celtic Warrior Sheamus looks to take back the World Heavyweight Championship from the man he lost it to, The World's Largest Athlete, The Big Show. And Mick Foley is back in WWE to lead five talented and eclectic Superstars into battle against the fearsome Team Ziggler when WWE presents Survivor Series!
Scorched 2008
In a climate change ravaged world in 2012, after 240 days without rain Sydney has only two weeks of water left. When the city is then ringed by severe bushfires, the question becomes, how do you fight fire when you have no water?
Drama | Science Fiction | Thriller
2012: Curse of the Xtabai 2012
A sealed ancient Mayan cave would be blown open unleashing Xtabai to wreak havoc on the world and destroy civilization. She begins by infecting the inhabitants of a nearby village with a terrible and deadly fever. The government responds by placing the village under quarantine. A team of students along with their teacher escape into the jungle to find a cure, guided only by a dream. They go deep into the bowels of the Mayan underworld, hunted by Xtabai herself. Will they ultimately succeed in defeating the evil Xtabai and saving mankind from her curse? How many will live to tell the tale?
Comedy | Horror | Thriller
Nostradamus: 2012 2009
Documentary about Nostradamus's quatrains.
Documentary | Drama | History
WWE Elimination Chamber 2012 2012
The Road to WrestleMania goes through one callous coliseum, the Elimination Chamber. Over the last decade, championships, careers & legacies have been at stake in the unpredictable environment, and this year will be no exception as CM Punk and Daniel Bryan will try to survive the sadistic structure with their respective titles in tact. Elimination Chamber (2012) was a PPV that took place on February 19, 2012 at the Bradley Center in Milwaukee, Wisconsin. The pay-per-view was the third annual Elimination Chamber event. Apart from the two Elimination Chamber matches, John Cena fought Kane in an Ambulance match, Beth Phoenix defended the Divas Championship against Tamina, and Jack Swagger defended the US Championship against Justin Gabriel.
London 2012 Olympic Opening Ceremony: Isles of Wonder 2012
The London 2012 Olympic Games Opening Ceremony took place at 9pm on 27 July 2012. Titled 'Isles of Wonder', the Ceremony welcomed the finest athletes from more than 200 nations for the start of the London 2012 Olympic Games, marking an historic third time the capital has hosted the world's biggest and most important sporting event. The Opening Ceremony reflected the key themes and priorities of the London 2012 Games, based on sport, inspiration, youth and urban transformation. It was a Ceremony 'for everyone' and celebrated contributions the UK has made to the world through innovation and revolution, as well as the creativity and exuberance of British people.
Hitchcock 2012
HITCHCOCK follows the relationship between director Alfred Hitchcock and his wife Alma Reville during the making of his most famous film, PSYCHO, and the trials and tribulations the director faced from Hollywood censors.
The Road to WrestleMania begins at the 25th Anniversary of the Royal Rumble, and 30 men look to realize a dream by winning the Royal Rumble match and headline WrestleMania. WWE Champion CM Punk faces long odds against retaining his championship when he deals with both #1 contender Dolph Ziggler and a special guest referee with a particular axe to grind, Interim Raw General Manager John Laurinaitis. Daniel Bryan looks to retain the World Heavyweight Championship, but he must overcome two colossal competitors, Mark Henry and Big Show in a steel cage. Plus John Cena and Kane square off in a bitter personal bout. The Scottrade Center in St. Louis, Missouri is this year's host for WWE Royal Rumble!
2012: Time for Change 2010
2012: Time For Change is a documentary feature that presents ways to transform our unsustainable society into a regenerative planetary culture. This can be achieved through a personal and global change of consciousness and the systemic implementation of ecological design.
Animation | Documentary
The Perfect Storm engulfs Los Angeles on the historic 25th Anniversary of SummerSlam. The wrecking ball force of Brock Lesnar meets the blind rage of a vengeful Triple H as two larger-than life gladiators collide for the first-time ever! Plus, CM Punk seeks validation for his status atop the WWE in a Triple Threat title defense against the Big Show and long-time rival John Cena. And after being injured at the hands of Alberto Del Rio, a determined Sheamus puts his World Heavyweight Championship on the line against the ruthless "Man of Destiny". A violent whirling of storm clouds approaches the Staples Center for WWE SummerSlam!
Everything I saw during 2012. This work will show you the dynamic moment of changing from film image to digital image. —Takashi Makino
WWE Extreme Rules 2012 2012
After eight years away from WWE, Brock Lesnar has reemerged to bring the pain, but John Cena refuses to be intimidated. CM Punk returns home to defend the WWE Championship and his family against Chris Jericho. Daniel Bryan faces Sheamus in a 2-out of-3 Falls Match! It's time to toss out the rulebook as WWE presents Extreme Rules! Extreme Rules (2012) took place on April 29, 2012 at the Allstate Arena in Rosemont, Illinois. It was the fourth event under the Extreme Rules chronology.
Defcon 2012 2010
On October 30, 2009 an independent filmmaker and his cast and crew are reported missing a month before the release of their controversial film predicting the end of the world. Authorities have reason to believe the film's 2012 doomsday scenario may have something to do with their disappearance. An agent for CEA (Central Extraterrestrial Agency) discovers the footage to the film and watches the film on September 10, 2012, three months before the world is to come to an end.
Science Fiction | Action
WWE Over The Limit 2012 2012
John Cena takes on the polarizing purveyor of "People Power", John Laurinaitis. Plus, both World Championships are on the line! The Superstars of Raw and SmackDown dial up the intensity as WWE goes Over The Limit! Over the Limit was a PPV which took place at the PNC Arena in Raleigh, North Carolina on May 20, 2012. It was the third and final event under the Over the Limit chronology. In a main event from the Raw brand, John Laurinaitis challenged John Cena. Another main event from the Raw brand featured CM Punk defending against Daniel Bryan for the WWE Championship. In the main event from the SmackDown brand, Sheamus defended against Alberto Del Rio, Randy Orton, and Chris Jericho in a Fatal Four-way match for the World Heavyweight Championship.
Ai no Kusabi 2012
This futuristic tale is set in a world where men are assigned various social classes based on their hair color. Iason Mink, a high-class Blondie, runs into Riki, a black-haired Mongrel, and makes him his "Pet." As Riki learns of the dangers Iason faces by keeping him, he finds himself developing feelings for his master. This is a remake of the 1992 OVA of the same name.
Drama | Animation | Science Fiction
Kingdom 2012
In the Warring States Period of ancient China (475-221 BCE), Shin and Hyou are war-orphans in the kingdom of Qin. They dream of one day proving themselves on the battlefield. One day, however, Hyou is taken to the palace by a minister. Winding up on the losing side of a power-struggle, Hyou manages to return to the village, barely alive. Shin then meets a boy who closely resembles Hyou, Ei Sei. For now he is the king of Qin; later he will become the emperor Shi Huangdi.
Action & Adventure | Animation | Drama
Star Awards 2012
Star Awards 2012 was a double television award ceremony held in Singapore. It is part of the annual Star Awards organised by MediaCorp for the two free-to-air channels, MediaCorp Channel 8 and MediaCorp Channel U. Star Awards 2012 was telecasted 'LIVE' on Channel 8, on 22 April 2012 and 29 April 2012. Both ceremonies commenced at 1900 hrs, where the results were announced. The 1st ceremony was held at the MediaCorp TV Theatre while the 2nd ceremony was held at Marina Bay Sands. In addition, Astro AEC also broadcast the ceremony 'live' on 29 April 2012. As with the previous two years, viewers was able to catch the presentation of the Professional awards on Channel 8. The ceremony was aired live on April 22, while the annual Star Awards prize presentation was aired on April 29. A post-show was held after the awards ceremony on April 29, which was telecasted on Channel U at 2200hrs.
Junior Dance 2012
Junior Dance is a program in the Netherlands. The hosts of this program are Ewout Genemans & Kim-Lian van der Meij who have presented the Junior Eurovision Song Contest 2012 in Amsterdam.
Fan Pan Tae 2012
แฟนพันธุ์แท้ 2012 เป็นรายการโทรทัศน์ไทย ผลิตโดยเวิร์คพอยท์ เอ็นเทอร์เทนเมนท์ เริ่มออกอากาศในวันที่ 3 กุมภาพันธ์ พ.ศ. 2555 ทางสถานีวิทยุโทรทัศน์กองทัพบกช่อง 5 เวลา 22:30-00:00 นาฬิกา มีกฤษณ์ ศรีภูมิเศรษฐ์ เป็นพิธีกร รายการนี้กลับมาอีกครั้งหลังจากที่ห่างหายไปนานกว่า 2 ปี ออกอากาศครั้งหลังสุดคือเมื่อ 17 เมษายน 2552 มีการแข่งขันทั้งหมด 38 เรื่อง เรื่องใหม่ 37 เรื่อง เรื่องเก่า 1 เรื่อง มีสุดยอดแฟนพันธุ์แท้ที่ตอบผิด 12 เรื่อง ตอบถูก 26 เรื่อง มีผู้เข้าแข่งขันสุดยอดแฟนพันธุ์แท้แห่งปี ทั้งหมด 25 คน จนวันที่ 2 พฤศจิกายน พ.ศ. 2555 คือ เทปพิเศษที่เปิดตัวผู้เข้าแข่งขัน แต่มีผู้เข้าแข่งขันสละสิทธิ์ไป 3 คน และมีกิจกรรมพิเศษให้สำหรับสุดยอดแฟนพันธุ์แท้ที่ตอบคำถามรอบสุดยอดแฟนพันธุ์แท้ผิด มีสิทธิ์เข้ากลับมาแข่งขันในการแข่งขัน สุดยอดแฟนพันธุ์แท้แห่งปี 2012 ได้อีกครั้ง โดยการเปิดโหวตผ่านทางเฟสบุ๊คของทางรายการ ผู้ที่มีผลโหวตมากที่สุดจะมีสิทธิ์เข้าร่วมแข่งขัน จึงมีผู้เข้าแข่งขันในปีนี้ทั้งหมด 23 คน อนึ่ง สิชฌพัฒน์ เจียรนัยอาภรณ์ สุดยอดแฟนพันธุ์แท้ 50 ปีโตโยต้า เป็นสุดยอดแฟนพันธุ์แท้คนแรกของปี 2012 ที่ตอบคำถามสุดยอดแฟนพันธุ์แท้ถูกแต่ไม่ได้เข้าร่วมการแข่งขันสุดยอดแฟนพันธุ์แท้แห่งปี 2012 เนื่องจากเป็นเทปพิเศษฉลองครบรอบ 50 ปี ได้รับรางวัลรถยนต์ Toyota Prius Hybrid แล้ว ในการค้นหาชิงชัยสุดยอดแฟนพันธุ์แท้แห่งปี 2012 เริ่มตั้งแต่วันที่ 16 พฤศจิกายน พ.ศ. 2555 และสิ้นสุดในวันที่ 1 มีนาคม พ.ศ. 2556 โดยสุดยอดแฟนพันธุ์แท้แห่งปี 2012 ได้แก่ เศกฤทธิ์ มะลิวัลย์ สุดยอดแฟนพันธุ์แท้กีตาร์
Soñando por Bailar 2012
Soñando por Bailar 2012 was the second season of Soñando por Bailar, an Argentinian reality show broadcast by El Trece. Unlike the first season, this time the contestants were professional dancers and every week they received a score from 1 to 10. The winner of the show got the chance to dance in Bailando 2012. The academy this season is placed in a castle in General Rodríguez. The show was hosted by Santiago del Moro and the jury are Marcelo Polino, Amalia Granata, Silvina Escudero, Reina Reech and Ángel de Brito. The season premiered was on December 3, 2011 and the finale was on May 6, 2012. Magdalena Bravi was the big winner.
86th Annual Macy's Thanksgiving Day Parade (2012)
First year in the lineup: Hello Kitty (second version), Papa Smurf, The Elf on the Shelf, Kaws's "Companion", Happy Dragon (second version), Novelty pumpkins (second version)
The Victoria's Secret Fashion Show 2012
This year's fashion show will be back in New York City with world-famous Victoria's Secret Angels Miranda Kerr, Alessandra Ambrosio, Adriana Lima, Doutzen Kroes, Candice Swanepoel, Erin Heatherton, Lily Aldridge, Lindsay Ellingson, and many more. The lingerie runway show will also include musical performances, pink carpet interviews, model profiles and a behind-the-scenes look at the making the fashion show.
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