text:
\section{Introduction}

One might say the history of intersection bodies began with the
paper of Busemann \cite{4}. Intersection bodies were first explicitly
defined and named by Lutwak \cite{11}. It was here that the duality
between intersection bodies and projection bodies was first made
clear. Despite considerable ingenuity of earlier attacks on
the Busemann--Petty problem, it seems fair to say that the work
of Lutwak \cite {11} represents the beginning of its eventual solution. In
\cite{11}, Lutwak also showed that if a convex body is sufficiently
smooth and not an intersection body, then there exists a centred
star body such that the conditions of Busemann--Petty problem
holds, but the result inequality is reversed. Following Lutwak,
the intersection body of order $i$ of a star body is introduced by
Zhang \cite{21}. It follows from this definition that every intersection
body of order $i$ of a star body is an intersection body of a star
body, and vice versa. As Zhang observes, the new definition of
intersection body allows a more appealing formulation, namely: the
Busemann--Petty problem has a positive answer in $n$-dimensional
Euclidean space if and only if each centered convex body is an
intersection body. The intersection body plays an essential role
in Busemann's theory \cite{5} of area in Minkowski spaces. The
intersection body is also an important matter of the
Brunn--Minkowski theory.

In recent years, some authors including Ball \cite{1,2}, Bourgain \cite{3},
Gardner \cite{6,7,8}, Schneider \cite{19} and Lutwak \cite{12,13,14,15,16,17,18} have given
considerable attention to the Brunn--Minkowski theory and their
various generalizations. The purpose of this paper is to
establish the Minkowski inequality for the dual quermassintegral
sum, which is a generalization of the Minkowski inequality for
mixed intersection bodies. Then, the Brunn--Minkowski inequality
and the Aleksandrov--Fenchel inequality for mixed intersection
bodies are proved and some related results are also given. In this
work we shall derive, for intersection bodies, all the analogous
inequalities for Lutwak's mixed projection body inequalities \cite{15}.
Thus, this work may be seen as presenting additional evidence of
the natural duality between intersection and projection bodies.

\section{Notation and preliminaries}

The setting for this paper is an $n$-dimensional Euclidean space
${\Bbb R}^{n} \ (n>2)$. Let ${\Bbb C}_{n}$ denote the set of
non-empty convex figures (compact, convex subsets) and let ${\cal
K}^{n}$ denote the subset of  ${\Bbb C}_{n}$ consisting of all
convex bodies (compact, convex subsets with non-empty interiors)
in ${\Bbb R}^n$. We reserve $u$ for unit vectors, and
$B$ for the unit ball centered at the
origin. The surface of $B$ is $S^{n-1}$. For $u\in S^{n-1}$, let
$E_u$ denote the hyperplane, through the origin, that is
orthogonal to $u$. Let $K^{u}$ to denote the image of $K$
under an orthogonal projection onto the hyperplane $E_u$. We use
$V(K)$ for the $n$-dimensional volume of convex body $K$. The
support function of $K\in {\cal K}^n$, $h(K,\cdot)$, defined on
${\Bbb R}^{n}$ by $h(K,\cdot)=\max\{x\cdot y: y\in K\}.$ Let
$\delta$ denote the Hausdorff metric on ${\cal K}^n$; i.e., for
$K, L\in {\cal K}^{n}$, $\delta(K,L)=|h_K-h_L|_{\infty}$, where
$|\cdot|_{\infty}$ denotes the sup-norm on the space of continuous
functions, $C(S^{n-1}).$

Associated with a compact subset $K$ of ${\Bbb R}^n$, which is
star-shaped with respect to the origin, its radial function
$\rho(K,\cdot): S^{n-1}\rightarrow {\Bbb R}$, defined for $u\in
S^{n-1}$, by $\rho(K,u)=\max\{\lambda\geq 0: \lambda u\in K\}$. If
$\rho(K,\cdot)$ is positive and continuous, $K$ will be called a
star body. Let $\varphi^{n}$ denote the set of star bodies in
${\Bbb R}^{n}$.\vspace{.5pc}

\subsection{\it Dual mixed volumes}

If $K_{1},\ldots,K_{r}\in \varphi^n$ and
$\lambda_{1},\ldots,\lambda_{r}\in {\Bbb R}$, then the radial
Minkowski  linear combination,
$\lambda_{1}K_{1}\tilde{+}\cdots\tilde{+}\lambda_{r} K_{r}$, is
defined by $\lambda_{1}K_{1}\tilde{+}\cdots\tilde{+}\lambda_{r}
K_{r}=\{\lambda_{1}x_{1}\tilde{+}\cdots\tilde{+}\lambda_{r} x_{r}$:
$x_{i}\in K_{i}\}$.

The following property will be used later. If $K, L\in \varphi^n$
and $\lambda, \mu\geq 0$,
\renewcommand\theequation{\thesubsection.\arabic{equation}}
\begin{equation}
\rho(\lambda K\tilde{+}\mu L,\cdot)=\lambda
\rho(K,\cdot)+\mu\rho(L,\cdot).
\end{equation}
For $K_{1},\ldots,K_{r}\in \varphi^n$ and
$\lambda_{1},\ldots,\lambda_{r}\geq 0$, the volume of the radial
Minkowski linear combination
$\lambda_{1}K_{1}\tilde{+}\cdots\tilde{+}\lambda_{r}K_{r}$ is a
homogeneous $n$th-degree polynomial in the $\lambda_{i}$ \cite{19},
\begin{equation}
V(\lambda_{1}K_{1}\tilde{+}\cdots\tilde{+}\lambda_{r}K_{r})=\sum \tilde{V}_{i_{1},\ldots,i_{n}}\lambda_{i_{1}}\cdots\lambda_{i_{n}},
\end{equation}
where the sum is taken over all $n$-tuples $(i_{1},\ldots,i_{n})$
whose entries are positive integers not exceeding $r$. If we
require the coefficients of the polynomial in (1.1.2) to be
symmetric in their arguments, then they are uniquely determined.
The coefficient $\tilde{V}_{i_{1},\ldots,i_{n}}$ is non-negative
and depends only on the bodies $K_{i_{1}},\ldots,K_{i_{n}}$. It is
written as $\tilde{V}(K_{i_{1}},\ldots,K_{i_{n}})$ and is called
the {\it dual mixed volume} of $K_{i_{1}},\ldots,K_{i_{n}}$. If
$K_{1}=\cdots=K_{n-i}=K$, $K_{n-i+1}=\cdots=K_{n}=L$, the dual
mixed volumes is written as $\tilde{V}_{i}(K,L)$ and the dual mixed
volumes $\tilde{V}_{i}(K,B)$ is written as $\tilde{W}_{i}(K)$.

For $K,L\in \varphi^n$ and $i\in R$, the $i$th dual mixed volume
of $K$ and $L$, $\tilde{V}_{i}(K,L)$, is defined by \cite{14}
\begin{equation*}
\tilde{V}_{i}(K,L)=\frac{1}{n}\int_{S^{n-1}}\rho(K,u)^{n-i}\rho(L,u)^{i}\hbox{d}S(u).
\end{equation*}
From the above identity, if $K\in \varphi^{n}$, $i\in {\Bbb R}$, then
\begin{equation}
\tilde{W}_{i}(K)=\frac{1}{n}\int_{S^{n-1}}\rho(K,u)^{n-i}\hbox{d}S(u).
\end{equation}

If $K_i\in \varphi^{n}\, (i=1,2,\ldots,n-1)$, then the dual mixed
volume of $K_{i}\cap E_{u}\, (i=1,2,\ldots,n-1)$ will be denoted by
$\tilde{v}(K_{1}\cap E_{u},\ldots,K_{n-1}\cap E_u)$. If
$K_{1}=\cdots=K_{n-1-i}=K$ and $K_{n-i}=\cdots=K_{n-1}=L,$ then
$\tilde{v}(K_{1}\cap E_{u},\ldots,K_{n-1}\cap E_{u})$ is written as
$\tilde{v}_{i}(K\cap E_{u},L\cap E_{u})$. If $L=B$, then
$\tilde{v}_{i}(K\cap E_ {u},B\cap E_{u})$ is written as
$\tilde{w}_{i}(K\cap E_{u})$.\vspace{.5pc}

\subsection{\it Intersection bodies}

For $K\in \varphi^n$, there is a unique star body $I\!K$ whose
radial function satisfies for $u\in S^{n-1}$,
\setcounter{equation}{0}
\begin{equation}
\rho(I\!K,u)=v(K\cap E_{u}).
\end{equation}
It is called the {\it intersection bodies} of $K$. From a result
of Busemann, it follows that $I\!K$ is a convex if $K$ is convex and
centrally symmetric with respect to the origin. Clearly any
intersection body is centred.

The volume of the intersection bodies is given by
$V(I\!K)=\frac{1}{n}\int_{S^{n-1}}v(K\cap E_{u})^{n}\hbox{d}S(u)$.

The mixed intersection bodies of $K_{1},\ldots,K_{n-1}\in
\varphi^{n}$, $I(K_{1},\ldots,K_{n-1})$, whose radial function is
defined by
\begin{equation}
\rho(I(K_{1},\ldots,K_{n-1}),u)=\tilde{v}(K_{1}\cap E_{u},\ldots,K_{n-1}\cap E_{u}),
\end{equation}
where $\tilde{v}$ is $(n-1)$-dimensional dual mixed volume.

If $K\in \varphi^{n}$ with $\rho(K,u)\in C(S^{n-1})$, and $i\in
{\Bbb R}$ is positive, the {\it intersection body of order i} of
$K$ is the centered star body $I_{i}K$ such that \cite{3}
$\rho(I_{i}K)=\frac{1}{n-1}\int_{S^{n-1}}\rho(K,u)^{n-i-1}\hbox{d}S(u)$,
for $u\in S^{n-1}$, where
$I_{i}K=I(\underbrace{K,\ldots,K}_{n-i-1},\underbrace{B,\ldots,B}_{i})$.

If $K_{1}=\cdots=K_{n-i-1}=K, K_{n-i}=\cdots=K_{n-1}=L$, then
$I(K_{1},\ldots,K_{n-1})$ is written as $I_{i}(K,L)$. If $L=B$,
then $I_{i}(K,L)$ is written as $I_{i}K$ and is called the $i$th
intersection body of $K$. For $I_{0}K$ simply write $I\!K$. The term
is introduced by Zhang \cite{21}.

The following properties will be used later: If $K, L$, $M,
K_{1},\ldots,K_{n-1}\in \varphi^{n}$ and $\lambda, \mu,
\lambda_{1},\ldots,\lambda_{n-1}>0$, then
\begin{align}
&I(\lambda K\tilde{+}\mu L, M) =\lambda I(K,M)\tilde{+}\mu I(L,M),\quad \hbox{where}\ M=(K_{1},\ldots,K_{n-2}).\\
&I(\lambda_{1}K_{1},\ldots,\lambda_{n-1}K_{n-1}) =\lambda_{1}\cdots\lambda_{n-1}I(K_{1},\ldots,K_{n-1}).
\end{align}

\section{Main results}

The following results will be required to prove our main Theorems.

\begin{lemm}
If $K,L\in \varphi^{n}${\rm ,} $0\leq i<n$ and $0<j<n-1${\rm ,} then
\begin{align*}
\tilde{W}_{i}(I\!K) &=\frac{1}{n}\int_{S^{n-1}}v(K\cap E_{u})^{n-i}{\rm d}S(u),\\[.5pc]
\tilde{W}_{i}(I_{j}K) &=\frac{1}{n}\int_{S^{n-1}}\tilde{w}_{j}(K\cap E_{u})^{n-i}{\rm d}S(u),
\end{align*}
\begin{equation*}
\tilde{W}_{i}(I_{j}(K,L)) =\frac{1}{n}\int_{S^{n-1}}\tilde{v}_{j}(K\cap E_{u},L\cap E_{u})^{n-i}{\rm d}S(u).
\end{equation*}
\end{lemm}

To prove this, we use (1.1.3) in conjunction with the fact (1.2.2).

\begin{lemm}
{\rm \cite{14}}. If $K_{1},\ldots,K_{n}\in \varphi^{n}${\rm ,} then
\begin{equation*}
\tilde{V}(K_{1},\ldots,K_{n})^{r}\leq \prod_{j=1}^{r}\tilde{V}(\underbrace{K_{j},\ldots,K_{j}}_{r},K_{r+1},\ldots,K_{n})
\end{equation*}
with equality if and only if $K_{1},\ldots,K_{n}$ are
all dilations of each other.
\end{lemm}

We shall need the following trivial elementary inequality.

\begin{lemm}
If $a, b\geq 0$ and $c, d>0${\rm ,} then for $0<p<1${\rm ,}
\begin{equation*}
(a+b)^{p}(c+d)^{1-p}\geq a^{p}c^{p-1}+b^{p}d^{p-1},
\end{equation*}
with equality if and only if $ad=bc$.
\end{lemm}

\begin{proof}
Consideration the following function
\begin{equation*}
f(x)=(x+b)^{p}(c+d)^{1-p}-x^{p}c^{1-p},\qquad x\geq 0.
\end{equation*}

Let $f'(x)=p(c+d)^{1-p}(x+b)^{p-1}-p c^{1-p}x^{p-1}=0$, we get
$x={bc}/{d}$. If $x\in (0,\frac{bc}{d})$, then $f'(x)<0$; if
$x\in \left(\frac{bc}{d},+\infty\right)$, then $f'(x)>0$. It follows that
\begin{equation*}
\min\limits_{x\geq 0}\{f(x)\}=f\left(\frac{bc}{d}\right)=b^{p}d^{1-p}.
\end{equation*}

This completes the proof.\hfill$\Box$\vspace{.6pc}
\end{proof}

\subsection{\it The Minkowski inequality for dual quermassintegral sum of mixed intersection bodies}

In \cite{10}, Leng introduce the concept of $i$-{\it quermassintegral
difference function} of convex bodies as follows: If $K,D\in {\cal
K}^{n}$ and $D\subset K$, then $i$-quermassintegral difference
function of convex bodies $K$ and $D$, $D_{w_{i}}(K,D)$, is
defined by
\begin{equation*}
D_{w_{i}}(K,D)=W_{i}(K)-W_{i}(D)\qquad (0\leq i\leq n-1).
\end{equation*}

In the section, we first introduce a new concept, {\it dual
quermassintegral sum function}, as follows:

If $K,D\in \varphi^{n}$, then the dual quermassintegral sum
function of star bodies $K$ and $D$, $S_{\tilde{w}_{i}}(K,D)$, is
defined by
\begin{equation*}
S_{\tilde{w}_{i}}(K,D)=\tilde{W}_{i}(K)+\tilde{W}_{i}(D)\qquad (0\leq i\leq n-1).
\end{equation*}
When $i=0$, we have $S_{v}(K,D)=V(K)+V(D)$, which is called the
{\it dual volume sum function} of star bodies $K$ and $L$.

The following Minkowski inequality for mixed intersection bodies
will be established: If $K, L\in \varphi^{n}$, and $0\leq i< n$
and $0<j<n-1$, then
\setcounter{equation}{-1}
\begin{equation}
\tilde{W}_{i}(I_{j}(K,L))^{n-1}\leq \tilde{W}_{i}(I\!K)^{n-j-1}
\tilde{W}_{i}(I\!L)^{j},
\end{equation}
with equality if and only if $K$ and $L$ are dilates.\pagebreak

This is just the special case $D=D'=\O$ of the following inequality.

\begin{theor}[\!]
If $K, L, D, D'\in \varphi^{n}$. Let $D'$ is a dilate copy of $D${\rm ,} and $0\leq i<
n$ and $0<j<n-1${\rm ,} then
\begin{equation}
S_{\tilde{w}_{i}}(I_{j}(K,L),I_{j}(D,D'))^{n-1}\leq S_{\tilde{w}_{i}}
(I\!K, I\!D)^{n-j-1}S_{\tilde{w}_{i}}(I\!L, I\!D')^{j},
\end{equation}
with equality if and only if $K$ and $L$ are dilates.
\end{theor}

\begin{proof}
In view of the special case of Lemma B, we obtain that
\begin{equation}
\tilde{v}_{j}(K\cap E_{u},L\cap E_{u})^{n-i}\leq v(K\cap E_{u})^{\frac{(n-i)(n-j-1)}{n-1}}v(L\cap E_{u})^{\frac{j(n-i)}{n-1}}
\end{equation}
with equality if and only if $K\cap E_{u}$ and $L\cap E_{u}$ are
dilates. It follows if and only if $K$ and $L$ are
dilates \cite{20}.

From Lemma A, eq. (2.1.2) and in view of Minkowski inequality for
integral \cite{9}, we have for $i<n-1$,
\begin{align}
n\tilde{W}_{i}(I_{j}(K,L)) &=\left(\|\tilde{v}_{j}(K\cap E_{u},L\cap E_{u})\|_{n-i}\right)^{n-i}\nonumber\\[.5pc]
&\leq \left(\|v(K\cap E_{u})^{\frac{n-j-1}{n-1}}v(L\cap E_{u})^{\frac{j}{n-1}}\|_{n-i}\right)^{n-i}\nonumber\\[.5pc]
&\leq \left(\|v(K\cap E_{u}\|_{n-i}\right)^{\frac{(n-i)(n-j-1)}{n-1}}\left(\|v(K\cap E_{u}\|_{n-i}\right)^{\frac{j(n-i)}{n-1}}\nonumber\\[.5pc]
&=(n\tilde{W}_{i}(I\!K))^{\frac{(n-j-1)}{n-1}}(n\tilde{W}_{i}(I\!L))^{\frac{j}{n-1}}\nonumber\\[.5pc]
&=n\tilde{W}_{i}(I\!K)^{\frac{(n-j-1)}{n-1}}\tilde{W}_{i}(I\!L)^{\frac{j}{n-1}}.
\end{align}

In view of the conditions of (2.1.2) and Minkowski inequality for
integral, it follows that the equality holds if and only if $K$
and $L$ are dilates.

Moreover, we consider the case of $i=n-1$ of the inequality
(2.1.3). If $i=n-1$, inequality (2.1.3) reduces to
\begin{equation*}
\tilde{W}_{n-1}(I_{j}(K,L))^{n-1}\leq \tilde{W}_{n-1}(I\!K)^{n-j-1}\tilde{W}_{n-1}(I\!L)^{j}.\tag{*}
\end{equation*}
From Lemma A, (*) changes to
\begin{align*}
&\left(\int_{S^{n-1}}\tilde{v}_{j}(K\cap E_{u},L\cap E_{u})\hbox{d}S(u)\right)^{n-1}\\[.5pc]
&\quad\, \leq \left(\int_{S^{n-1}}v(K\cap E_{u})
\hbox{d}S(u)\right)^{n-j-1}\left(\int_{S^{n-1}}v(L\cap
E_{u})\hbox{d}S(u)\right)^{j}.\tag{**}
\end{align*}
On the other hand, integrating
both sides of (2.1.2) and in view of H\"{o}lder inequality for
integral, we obtain
\begin{align*}
&\int_{S^{n-1}}\tilde{v}_{j}(K\cap E_{u},L\cap E_{u})\hbox{d}S(u)\\[.5pc]
&\qquad\, \leq
\int_{S^{n-1}}v(K\cap E_{u})^{\frac{n-j-1}{n-1}}v(L\cap E_{u})^{\frac{j}{n-1}}\hbox{d}S(u)\\[.5pc]
&\qquad\, \leq  \left(\int_{S^{n-1}}v(K\cap
E_{u})\hbox{d}S(u)\right)^{\frac{n-j-1}{n-1}}\left(\int_{S^{n-1}}v(L\cap
E_{u})\hbox{d}S(u)\right)^{\frac{j}{n-1}}.
\end{align*}

Moreover, from inequality (2.1.3), we obtain
\begin{equation*}
\tilde{W}_{i}(I_{j}(K,L))^{n-1}\leq \tilde{W}_{i}(I\!K)^{n-j-1}\tilde{W}_{i}(I\!L)^{j},
\end{equation*}
with equality if and only if $K$ and $L$ are dilates, and
\begin{equation*}
\tilde{W}_{i}(I_{j}(D,D'))^{n-1}=\tilde{W}_{i}(I\!D)^{n-j-1}
\tilde{W}_{i}(I\!D')^{j}.
\end{equation*}
Hence, from the inequality in Lemma C, we have
\begin{align*}
S_{\tilde{w}_{i}}(I_{j}(K,L),I_{j}(D,D')) &\leq \tilde{W}_{i}(I\!K)^{(n-j-1)/(n-1)}
\tilde{W}_{i}(I\!L)^{j/(n-1)}\\[.3pc]
 &\quad\, +\tilde{W}_{i}(I\!D)^{(n-j-1)/(n-1)}\tilde{W}_{i}(I\!D')^{j/(n-1)}\\[.3pc]
&\leq S_{\tilde{w}}(I\!K, I\!D)^{n-j-1}S_{\tilde{w}}(I\!L, I\!D')^{j}.
\end{align*}
The proof of Theorem 2.1.1 is complete.\hfill$\Box$
\end{proof}

\setcounter{subsubsection}{0}
\begin{rem}
{\rm Taking $D=D'=\O$ and $j=1$ to (2.1.1), (2.1.1) changes to
\begin{equation*}
\tilde{W}_{i}(I_{1}(K,L))^{n-1}\leq \tilde{W}_{i}(I\!K)^{n-2}\tilde{W}_{i}
(I\!L),
\end{equation*}
with equality if and only if $K$ and $L$ are dilates.

This is just a dual form of the following inequality which was
given by Lutwak \cite{15}.}
\end{rem}

{\it The Minkowski inequality for mixed projection bodies}. If
$K, L\in {\cal K}^{n}$, and $0\leq i< n$, then
\begin{equation*}
W_{i}(\Pi_{1}(K,L))^{n-1}\geq W_{i}(\Pi K)^{n-2}W_{i}(\Pi L),
\end{equation*}
with equality if and only if $K$ and $L$ are homothetic.

A somewhat surprising consequence of Theorem 2.1.1 is the
following version.

\begin{theor}[\!]
If $K,L\in \eta\subset\varphi^{n}${\rm ,}
and $0\leq i<n${\rm ,} while $0<j<n-1$ and if either
\begin{equation}
\tilde{W}_{i}(I_{j}(K,M))=\tilde{W}_{i}(I_{j}(L,M)),\quad \hbox{for all}\ M\in \eta
\end{equation}
or
\begin{equation}
\tilde{W}_{i}(I_{j}(M,K))=\tilde{W}_{i}(I_{j}(M,L)),\quad \hbox{for all}\ M\in \eta
\end{equation}
hold{\rm ,} then it follows that $K=L${\rm ,} up to translation.
\end{theor}

\begin{proof}
Suppose (2.1.4) holds. Take $K$ for $M$ in
(2.1.4) and use the inequality (2.1.1). We obtain
\begin{equation*}
\tilde{W}_{i}(I\!K)=\tilde{W}_{i}(I_{j}(L,K))\leq \tilde{W}_{i}(I\!L)^{\frac{(n-j-1)}{n-1}}
\tilde{W}_{i}(I\!K)^{\frac{j}{n-1}}
\end{equation*}
with equality if and only if $K$ is a dilation of $L$.

Hence
\begin{equation*}
\tilde{W}_{i}(I\!K)\leq\tilde{W}_{i}(I\!L)
\end{equation*}
with equality if and only if $K$ is a dilation of $L$.

Similarly, take $L$ for $M$ in (2.1.4) and use again the
inequality (2.1.1). We get
\begin{equation*}
\tilde{W}_{i}(I\!K)\geq \tilde{W}_{i}(I\!L),
\end{equation*}
with equality if and only if $K$ is a dilation of $L$.

Hence
\begin{equation*}
\tilde{W}_{i}(I\!K)= \tilde{W}_{i}(I\!L)
\end{equation*}
and $K$ is a dilation of $L$. In view of the fact that the intersection bodies are
centered, there~exist $\lambda>0$ such that $K=\lambda L$,
and $\lambda^{(n-1)(n-i)}=1$, for $0\leq i<n-1$. Therefore
$\lambda=1$.~\hfill$\Box$
\end{proof}

Similar sort of argument shows that condition (2.1.5)
implies that $K$ and $L$ must be translates.

\setcounter{subsubsection}{1}
\begin{rem}
{\rm Theorem 2.1.2 is just the dual form of the
following `Theorem 5.4' which was given by Lutwak \cite{15}.}
\end{rem}

\setcounter{section}{5}
\setcounter{defin}{3}
\begin{theorem}[\!]
If $K,L\in \gamma\subset {\cal K}^{n}${\rm ,}
and $0\leq i<n${\rm ,} while $0<j<n-1$ and if either
\begin{equation*}
W_{i}(\Pi_{j}(K,M))=W_{i}(\Pi_{j}(L,M)),\quad \hbox{for}\ M\in \gamma
\end{equation*}
or
\begin{equation*}
W_{i}(\Pi_{j}(M,K))=W_{i}(\Pi_{j}(M,L)),\quad \hbox{for}\ M\in \gamma
\end{equation*}
hold{\rm ,} then it follows that $K=L${\rm ,} up to translation.
\end{theorem}

\setcounter{section}{2}
\subsection{\it The Aleksandrov--Fenchel inequality for mixed intersection bodies}

The Aleksandrov--Fenchel inequality for mixed intersection bodies
is as follows: If $K_{1},\ldots,K_{n-1}\in \varphi^{n}$, then
\begin{equation*}
V(I(K_{1},\ldots,K_{n-1}))\leq \prod_{j=1}^{r}V(I(\underbrace{K_{j},\ldots,K_{j}}_{r},K_{r+1},\ldots,K_{n-1}))
\end{equation*}
with equality if and only if $K_{1},\ldots,K_{n-1}$ are all
dilations of each other.

This is just the special case $i=0$ of the following.

\begin{theor}[\!]
If $K_{1},\ldots,K_{n-1}\in
\varphi^{n}${\rm ,} $0\leq i< n, 0<j<n-1$ and $0< r \leq n-1${\rm ,} then
\setcounter{equation}{0}
\begin{equation}
\tilde{W}_{i}(I(K_{1},\ldots,K_{n-1}))^{r}\leq\prod_{j=1}^{r}\tilde{W}_{i}(I(\underbrace{K_{j},\ldots,K_{j}}_{r},K_{r+1},\ldots,K_{n-1}))
\end{equation}
with equality if and only if
$K_{1},\ldots,K_{n-1}$ are all dilations of each other.
\end{theor}

\begin{proof}
When $i=1$, inequality (2.2.1) reduces to the inequality
in Lemma B. In the following, we suppose that $i<n-1$.\pagebreak

From (1.1.3) and (1.2.2), we have that
\begin{equation}
\tilde{W}_{i}(I(K_{1},\ldots,K_{n-1}))=\frac{1}{n}\int_{S^{n-1}}
\tilde{v}(K_{1}\cap E_{u},\ldots,K_{n-1}\cap E_{u})^{n-i}\hbox{d}S(u).
\end{equation}
By using the inequality in Lemma B, we easily get that
\begin{align}
&\tilde{v}(K_{1}\cap E_{u},\ldots,K_{n-1}\cap E_{u})^{n-i}\nonumber\\
&\quad\, \leq\left(\prod_{j=1}^{r}\tilde{v}(\underbrace{K_{j}\cap E_{u},\ldots,K_{j}\cap
E_{u}}_{r},K_{r+1}\cap E_{u},\ldots,K_{n-1}\cap
E_{u})\right)^{\frac{n-i}{r}},
\end{align}
with equality if and only if $K_{1}\cap E_{u},\ldots,K_{n-1}\cap E_{u}$ are all
dilations of each other. It follows if and only if
$K_{1},\ldots,K_{n-1}$ are all dilations of each other.

On the other hand, the H\"{o}lder's inequality can be stated as \cite{9}
\begin{equation}
\int_{S^{n-1}}\prod_{i=1}^{m}f_{i}(u)\hbox{d}S(u)\leq\prod_{i=1}^{m}\left(\int_{S^{n-1}}(f_{i}(u))^{m}\hbox{d}S(u)\right)^{1/m},
\end{equation}
with equality if and only if all $f_{i}$ are proportional.

From (2.2.2), (2.2.3) and (2.2.4), we obtain that

\begin{align*}
\tilde{W}_{i}(I(K_{1},\ldots,K_{n-1})) &=\frac{1}{n}\int_{S^{(n-1)}}\tilde{v}(K_{1}\cap E_{u},\ldots,K_{n-1}\cap E_{u})^{n-i}\hbox{d}S(u)\\
&\leq\frac{1}{n}\int_{S^{u}}\left(\prod_{j=1}^{r}\tilde{v}(\underbrace{K_{j}\cap E_{u},\ldots,K_{j}\cap E_{u}
}_{r},\right.\\
&\left.\quad\, \begin{array}{l} \\[2pc] \end{array}K_{r+1}\cap E_{u},\ldots,K_{n-1}\cap
E_{u})\right)^{\frac{n-i}{r}}\hbox{d}S(u)\\
&\leq \left(\prod_{j=1}^{r}\frac{1}{n}\int_{S^{n-1}}\tilde{v}(\underbrace{K_{j}\cap E_{u},\ldots,K_{j}\cap E_{u}
}_{r},\right.\\
&\quad\,\left. \begin{array}{l} \\[2pc] \end{array}K_{r+1}\cap E_{u},\ldots,K_{n-1}\cap
E_{u})^{n-i}\hbox{d}S(u)\right)^{r}\\
&=\left(\prod_{j=1}^{r}\tilde{W}_{i}(I\underbrace{(K_{j},\ldots,K_{j}}_{r},K_{r+1},\ldots,K_{n-1}))\right)^{r}.
\end{align*}

In view of the equality conditions (2.2.3) and (2.2.4), it
follows that the equality holds if and only if
$K_{1},\ldots,K_{n-1}$ are all dilations of each other.

The proof is complete.\hfill$\Box$
\end{proof}

\setcounter{subsubsection}{0}
\begin{rem}
{\rm The inequality (2.2.1) is just a dual form of
the following inequality which was given by Lutwak \cite{15}.}
\end{rem}

\noindent {\it The Aleksandrov--Fenchel inequality for mixed projection
bodies}. If $K_{1},\ldots,K_{n-1}\in {\cal K}^{n}$, $0\leq i<n,
1<j<n-1$ and $0< r \leq n-1$ then
\begin{equation*}
W_{i}(\Pi(K_{1},\ldots,K_{n-1}))^{r}\geq\prod_{j=1}^{r}W_{i}(\Pi(\underbrace{K_{j},\ldots,K_{j}}_{r},K_{r+1},\ldots,K_{n-1})).
\end{equation*}

From the case $r=n-1$ of inequality (2.2.1), it is as follows.

\setcounter{subsubsection}{0}
\begin{coro}$\left.\right.$\vspace{.5pc}

\noindent If $K_{1},\ldots,K_{n-1}\in
\varphi^{n}$, and $0\leq i< n$ then
\begin{equation}
\tilde{W}_{i}(I(K_{1},\ldots,K_{n-1}))^{n-1}\leq \tilde{W}_{i}
(I\!K_{1})\cdots\tilde{W}_{i}(I\!K_{n-1}),
\end{equation}
with equality if and only if $K_{1},\ldots,K_{n-1}$ are all
dilations of each other.
\end{coro}

\begin{rem}
{\rm Corollary (2.2.1) is similar to the following
`Theorem 5.2' which was given by Lutwak \cite{15}.}
\end{rem}

\setcounter{section}{5}
\setcounter{defin}{1}
\begin{theorem}[\!]
If $K_{1},\ldots,K_{n-1}\in {\cal K}^{n}$, and $0\leq i< n$ then
\begin{equation*}
W_{i}(\Pi(K_{1},\ldots,K_{n-1}))^{n-1}\geq W_{i}(\Pi
K_{1})\cdots W_{i}(\Pi K_{n-1}),
\end{equation*}
with equality if and only if $K_{1},\ldots,K_{n-1}$ are homothetic.
\end{theorem}

Taking $K_{1}=\cdots=K_{n-j-1}=K$ and $K_{n-j}=\cdots=K_{n-1}=L$
to (2.2.5), (2.2.5) reduces to (2.1.0). Taking
$K_{1}=\cdots=K_{r}=K$, $K_{r}=L$, and $K_{r+1}=\cdots=K_{n-1}=B$
to (2.2.1), (2.2.1) changes to the following.

\setcounter{section}{2}
\setcounter{subsubsection}{1}
\begin{coro}$\left.\right.$\vspace{.5pc}

\noindent If $K, L\in \varphi^{n}${\rm ,} and $0\leq i< n$ and $0\leq j<n-1${\rm ,} then
\begin{equation*}
\tilde{W}_{i}(I(\underbrace{K,\ldots,K}_{n-j-2},\underbrace{B,\ldots,B}_{j},L))^{n-j-1}\leq \tilde{W}_{i}(I_{j}K)^{n-j-2}\tilde{W}_{i}(I_{j}L),
\end{equation*}
with equality if and only if $K$ and $L$ are dilates.
\end{coro}

A somewhat surprising consequence of Corollary 2.2.2 is the
following version for mixed intersection bodies.

\setcounter{subsubsection}{1}
\begin{theor}[\!]
If $K,L\in \eta\subset \varphi^{n}${\rm ,}
$0\leq i<n-1, 0\leq j<n-1$ and if either
\begin{align}
&\tilde{W}_{i}(I(\underbrace{K,\ldots,K}_{n-j-1},\underbrace{B,\ldots,B}_{j},M))\nonumber\\[.5pc]
&\qquad\, =\tilde{W}_{i}(I(\underbrace{L,\ldots,L}_{n-j-1},\underbrace{B,\ldots,B}_{j},M)),\qquad
\hbox{for all}\ M\in \eta,
\end{align}
or
\begin{align}
&\tilde{W}_{i}(I(\underbrace{M,\ldots,M}_{n-j-1},\underbrace{B,\ldots,B}_{j},K))\nonumber\\[.5pc]
&\qquad\,=\tilde{W}_{i}(I(\underbrace{M,\ldots,M}_{n-j-1},\underbrace{B,\ldots,B}_{j},L)),\qquad\hbox{for all}\ M\in \eta,
\end{align}
hold{\rm ,} then it follows that $K$=$L${\rm ,} up to translation.
\end{theor}

\begin{proof}
Suppose (2.2.6) holds, take $K$ for $M$, use
Corollary~2.2.2, and get
\begin{align*}
\tilde{W}_{i}(I_{j}K)&=\tilde{W}_{i}(I(\underbrace{L,\ldots,L}_{n-j-1},\underbrace{B,\ldots,B}_{j},K))\\[.5pc]
&\leq \tilde{W}_{i}(I_{j}L)^{\frac{n-j-2}{n-j-1}}\tilde{W}_{i}(I_{j}K)^{\frac{1}{n-j-1}},
\end{align*}
with equality if and only if $K$ and $L$ are dilates.

Hence
\begin{equation*}
\tilde{W}_{i}(I_{j}K)\leq \tilde{W}_{i}(I_{j}L),
\end{equation*}
with equality if and only if $K$ and $L$ are dilates.

On the other hand, take $L$ for $M$, use Corollary 2.2.2 again,
and get
\begin{equation*}
\tilde{W}_{i}(I_{j}K)\geq \tilde{W}_{i}(I_{j}L),
\end{equation*}
with equality if and only if $K$ and $L$ are dilates.

Therefore
\begin{equation*}
\tilde{W}_{i}(I_{j}K)= \tilde{W}_{i}(I_{j}L),
\end{equation*}
where $K$ and $L$ are dilates and in view of the fact that the intersection bodies
are centered, there exists $\lambda>0$ such that $K=\lambda
L$. From (1.2.4), we have $\lambda^{(n-j-1)(n-i)}
\tilde{W}_{i}(I_{j}L)= \tilde{W}_{i}(I_{j}L),$ hence $\lambda=1$.

Similar argument shows that condition (2.2.7)
implies $K$=$L$, up to tanslation.\hfill$\Box$
\end{proof}

\begin{rem}
{\rm Taking $j=0$ to Theorem 2.2.2, it reduces to the
following:}\vspace{.6pc}

If $K,L\in \eta\subset \varphi^{n}$, $0\leq i<n$ and if either
\begin{equation*}
\tilde{W}_{i}(I_{1}(K,M))=\tilde{W}_{i}(I_{1}(L,M)),\qquad\hbox{for all}\ M\in \eta
\end{equation*}
or
\begin{equation*}
\tilde{W}_{i}(I_{1}(M,K))=\tilde{W}_{i}(I_{1}(M,L)),\qquad\hbox{for all}\ M\in \eta
\end{equation*}
hold, then it follows that $K=L$, up to translation.
\end{rem}

This is just the special case $j=1$ of Theorem~2.1.2.

\subsection{\it The Brunn--Minkowski inequality for mixed intersection bodies}

The Brunn--Minkowski inequality for intersection bodies, which will
be established is: If $K,L\in \varphi^{n}$, then
\begin{equation*}
V(I(K\tilde{+}L))^{1/n(n-1)}
\leq V(I\!K)^{1/n(n-1)}+V(I\!L)^{1/n(n-1)},
\end{equation*}
with equality if and only if $K$ and $L$ are dilates.

This is just the special case $i=0$ and $\alpha=1$ of the following.

\begin{theor}[\!]
If $K,L\in \varphi^{n}${\rm ,} and
$0\leq i<n${\rm ,} then for $0\leq \alpha \leq 1${\rm ,}
\setcounter{equation}{0}
\begin{align}
\tilde{W}_{i}(I(K\tilde{+}L))^{1/(n-i)(n-1)} &\leq
\tilde{W}_{i}(I(\alpha
K\tilde{+}(1-\alpha)L))^{1/(n-i)(n-1)}\nonumber\\
&\quad\, +\tilde{W}_{i}(I((1-\alpha)
K\tilde{+}\alpha L))^{1/(n-i)(n-1)},
\end{align}
with equality if and only if $(\alpha K\tilde{+}(1-\alpha)L)$ and
$(1-\alpha) K\tilde{+}\alpha L$ are dilates.
\end{theor}

\begin{proof}
Let $M=(L_{1},\ldots,L_{n-2})$, from (1.1.1), (1.1.3), (1.2.3)
and in view of the Minkowski inequality for
integral \cite{9}, we obtain that
\begin{align}
\tilde{W}_{i}(I(K\tilde{+}L,M))^{1/(n-i)} &=n^{-1/(n-i)}\|\rho(I(K\tilde{+}L,M),u)\|_{n-i}\nonumber\\[.3pc]
&=n^{-1/(n-i)}\|\rho(I(K,M)\tilde{+}I(L,M),u)\|_{n-i}\nonumber\\[.3pc]
&=n^{-1/(n-i)}\|\rho(I(K,M),u)+\rho(I(L,M),u)\|_{n-i}\nonumber\\[.3pc]
&\leq n^{-1/(n-i)}\left(\|\alpha\rho(I(K,M),u)\right.\nonumber\\
&\quad\,+(1-\alpha)\rho(I(L,M),u)\|_{n-i}+\|(1-\alpha)\nonumber\\
&\quad\,\left.\times\rho(I(K,M),u)+\alpha\rho(I(L,M),u)\|_{n-i}\right)\nonumber\\[.3pc]
&=n^{-1/(n-i)}\left(\|\rho(\alpha\cdot I(K,M)\tilde{+}(1-\alpha)\right.\nonumber\\
&\quad\, \times I(L,M),u)\|_{n-i}+\|\rho((1-\alpha)\cdot I(K,M)\nonumber\\
&\quad\,\left.\tilde{+}\alpha I(L,M),u)\|_{n-i}\right)\nonumber\\[.3pc]
&=n^{-1/(n-i)}\left(\|\rho(I(\alpha\cdot K\tilde{+}(1-\alpha)L,M),u)\|_{n-i}\right.\nonumber\\
&\quad\, \left.+\|\rho(I((1-\alpha)\cdot K\tilde{+}\alpha L,M),u)\|_{n-i}\right)\nonumber\\[.3pc]
&=\tilde{W}_{i}(I(\alpha\cdot K\tilde{+}(1-\alpha)L,M))^{1/(n-i)}\nonumber\\
&\quad\, +\tilde{W}_{i}(I(1-\alpha)\cdot K\tilde{+}\alpha L,M))^{1/(n-i)}.
\end{align}

On the other hand, taking $L_{1}=\cdots=L_{n-2}=K\tilde{+}L$ to
(2.3.2) and apply the inequality (2.1.0) twice, we get
\begin{align}
\tilde{W}_{i}(I(K\tilde{+}L))^{1/(n-i)}&\leq
\tilde{W}_{i}(I_{n-2}(\alpha\cdot
K\tilde{+}(1-\alpha)L,K\tilde{+}L))^{1/(n-i)}\nonumber\\
&\quad\, +\tilde{W}_{i}(I_{n-2}((1-\alpha)\cdot
K\tilde{+}\alpha L,K\tilde{+}L))^{1/(n-i)}\nonumber\\[.5pc]
&\leq \tilde{W}_{i}(I(\alpha\cdot
K\tilde{+}(1-\alpha)L))^{1/(n-1)(n-i)}\nonumber\\
&\quad\, \times\tilde{W}_{i}(I(K\tilde{+}L))^{(n-2)/(n-1)(n-i)}\nonumber\\
&\quad\,+\tilde{W}_{i}(I((1-\alpha)\cdot K\tilde{+}\alpha
L))^{1/(n-1)(n-i)}\nonumber\\
&\quad\, \times \tilde{W}_{i}(I(K\tilde{+}L))^{(n-2)/(n-1)(n-i)},
\end{align}
with equality if and only if $\alpha\cdot
K\tilde{+}(1-\alpha)L$, $(1-\alpha)\cdot K\tilde{+}\alpha L$ and
$M=K\tilde{+}L$ are dilates. Combining this with the equality
condition of (2.3.2), it follows that the condition holds if and
only if $K$ and $L$ are dilates.

Dividing both sides of (2.3.3) by
$\tilde{W}_{i}(I(K\tilde{+}L))^{(n-2)/(n-1)(n-i)}$, we get the
inequality (2.3.1).

The proof is complete.\hfill$\Box$
\end{proof}

Taking $\alpha=1$ in inequality (2.3.1), we have the following.

\setcounter{subsubsection}{0}
\begin{coro}$\left.\right.$\vspace{.5pc}

\noindent If $K,L\in \varphi^{n}${\rm ,} and $0\leq i<n${\rm ,} then
\begin{equation}
\tilde{W}_{i}(I(K\tilde{+}L))^{1/(n-i)(n-1)}\leq
\tilde{W}_{i}(I\!K)^{1/(n-i)(n-1)}+\tilde{W}_{i}(I\!L)^{1/(n-i)(n-1)},
\end{equation}
with equality if and only if $K$ and $L$ are dilates.
\end{coro}

\setcounter{subsubsection}{0}
\begin{rem}
{\rm From inequalities (2.3.4) and (1.2.4), we obtain that
\begin{align}
\tilde{W}_{i}(I(\alpha
K\tilde{+}(1-\alpha)L))^{1/(n-i)(n-1)}&\leq \alpha \tilde{W}_{i}(I\!K)^{1/(n-i)(n-1)}\nonumber\\
&\quad\, +(1-\alpha)\tilde{W}_{i}(I\!L)^{1/(n-i)(n-1)}
\end{align}
and
\begin{align}
\tilde{W}_{i}(I((1-\alpha)
K\tilde{+}\alpha L))^{1/(n-i)(n-1)}&\leq (1-\alpha) \tilde{W}_{i}(I\!K )^{1/(n-i)(n-1)}\nonumber\\
&\quad\, +\alpha\tilde{W}_{i}(I\!L)^{1/(n-i)(n-1)}.
\end{align}

From (2.3.5) and (2.3.6), we obtain that
\begin{align*}
&\tilde{W}_{i}(I(\alpha
K\tilde{+}(1-\alpha)L))^{1/(n-i)(n-1)}+\tilde{W}_{i}(I((1-\alpha)
K\tilde{+}\alpha L))^{1/(n-i)(n-1)}\\
&\quad\, \leq \tilde{W}_{i}(I\!K)^{1/(n-i)(n-1)}
+\tilde{W}_{i}(I\!L)^{1/(n-i)(n-1)}.
\end{align*}

This shows that inequality (2.3.1) is a strengthened form of
inequality (2.3.4).}
\end{rem}

\begin{rem}
{\rm Inequality (2.3.4) is just a dual form of
the following inequality which was given by Lutwak \cite{15}.}
\end{rem}

\noindent{\it The Brunn--Minkowski inequality for mixed projection bodies.}
If $K,L\in {\cal K}^{n}$, and $0\leq i<n,$ then
\begin{equation*}
W_{i}(\Pi(K+L))^{1/(n-i)(n-1)}\geq
W_{i}(\Pi K)^{1/(n-i)(n-1)}+W_{i}(\Pi L)^{1/(n-i)(n-1)},
\end{equation*}
with equality if and only if $K$ and $L$ are homothetic.

\section*{Acknowledgements}

This work was supported by the National Natural Sciences Foundation of China
(10271071), and by Academic Mainstay of Middle-age and Youth Foundation
of Shandong Province of China.

redpajama_set_name: RedPajamaArXiv


text:
2015 Guard Franklin Howard to Syracuse
Franklin Howard, a 6-foot-4 2015 combo guard from Fairfax (Va.) Paul VI and Team Takeover, committed to Syracuse on his 17th birthday.
Howard also considered Maryland, Georgetown, Ohio State, Virginia and N.C. State.
He missed this past season after undergoing ACL surgery last summer.
"Frank is on his way to full recovery from ACL last July," Paul VI coach Glen Farello told SNY.tv. "Frank's got the combination of skill, IQ and competitiveness that coaches appreciate. The kid is a winner. And he will be even more hungry coming back.
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He is Syracuse's third 2015 pledge, following Trenton (N.J.) Catholic guard Malachi Richardson and New Hampton (N.H.) power forward Tyler Lydon.
And like ZAGS on Facebook
Michigan State's Gary Harris Will Enter the NBA Draft
7-Footer Okoroh Changes Twitter Commitment to Cal from Tennessee After Cuonzo Martin Is Hired

redpajama_set_name: RedPajamaCommonCrawl


text:
Be aware of fine zones in London says FTA
The FTA has said that drivers are having a difficulty knowing how long to stay parked when unloading for deliveries, it has emerged.
There are 33 London boroughs in London all applying different rules for loading and unloading, usually dependant on the size of the vehicle and the time of day the delivery is being made, in many cases there is no clear signage to account for this. The Freight Transport Association has said drivers are unsure of the rules of what they can do in each area causing a great amount of confusion.
The FTA also claims that there has been a 50% leap in the number of penalty charge notices being issued, urging companies to train their drivers in identifying potential hotspots for fines.
Natalie Chapman, FTAs head of policy for London commented ' One of the issues with yellow lines for delivery drivers is that there are no signs to indicated exactly how much time is permitted for loading and unloading, and as this changes to borough to borough throughout London, drivers can't possibly be expected to know how long they have to make their deliveries'.
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text:
What's next after Election Day: Outstanding ballots, canvassing, and recounts
WHQR | By Ben Schachtman
Published November 5, 2020 at 4:24 PM EST
Benjamin Schachtman
The last votes were cast on, or postmarked by, Election Day. Now what?
Election Day has come and gone, but there's still plenty left unresolved. In North Carolina, over 100,000 absentee ballots remain 'outstanding,' including thousands in New Hanover County. Over the next week, the final tallies will continue to change as absentee and provisional ballots are evaluated, and the potential for recount request remains on the table.
The 2020 election saw record breaking turnout, but the results published by the North Carolina State Board of Elections (NCSBE) on Tuesday evening aren't the end of the story.
The state is currently waiting on up to 116,200 outstanding absentee ballots. That number could be lower: the total includes all requested absentee ballots that have not yet been received, but some residents who requested ballots may have instead voted in person --- or not at all. The final tally will include all absentee ballots received by November 12 and postmarked on November 3 or earlier.
The outstanding ballots skew towards unaffiliated and Democratic voters (45,300 unaffiliated, 43,000 Democratic, and 26,500 Republican voters). Locally, outstanding ballots are expected to fall along similar lines.
Locally, the NCSBE expects 3,200 absentee ballots in New Hanover County, 1,400 in Brunswick County, and 600 in Pender County.
It's worth noting that New Hanover County, which has a higher proportion of Democratic voters, has a higher percentage of outstanding absentee ballots (about 50% more absentee ballots per registered voter versus neighboring, more Republican-leaning counties). That's correlation, not causation --- but it lends credence to expectations that absentee ballots could lean more Democratic in New Hanover County.
There are also 40,766 provisional ballots state-wide, according to NCSBE, including 1,200 in New Hanover County, 869 in Brunswick County, and 340 in Pender County. These ballots are cast "when an individual's name does not appear on the poll book or other questions arise about that person's eligibility to vote or to vote a particular ballot," according to NCSBE.
The eligibility of provisional voters is being evaluated by county election boards; eligible ballots will be added to the final count along with absentee ballots.
The total numbers, including early, one-stop, absentee, provisional, and Election Day ballots won't be known until late next week, most likely after county board canvasses, which are scheduled for 11 a.m. Friday, November 13.
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That margin depends on the vote totals --- 1% of the total votes cast for two candidates involved in the challenge.
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In races under the supervision of county boards of election, candidates have until 5 p.m. on Monday, November 16. State level candidates have until noon on Tuesday, November 17.
Later this month, the State Board of Elections will certify statewide results for all federal, statewide, multi-district and judicial contests at a public meeting held at 11 a.m. Tuesday, Nov. 24. Updates from the NCSBE can be found here.
Local 2020 ElectionvotecanvassrecountAbsentee Ballots


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Home Featured Perry Co. walk organizer sees himself as part of problem and solution to area's health woes
Perry Co. walk organizer sees himself as part of problem and solution to area's health woes
Sam Neace, organizer of the Great Perry County Walk Saturday, Sept. 20, knows firsthand that healthy habits begin with a single step. (Photo by Cris Ritchie)
By Kristy Robinson Horine
KyForward correspondent
When it comes to health and fitness, Sam Neace might be one of the worst role models around. At least, that's the way he sees it.
"I don't pay anywhere near as much attention to how healthy I am eating, and I don't exercise as much as I need to," says Neace. "I have a real bad habit, if I don't have something motivating me, I fall behind."
That self-assessment makes Neace human, and relatively normal. What makes this life-long resident of Perry County extraordinary is how he has decided to take steps in the right direction – about 22,000 steps, that is.
On Saturday, Sept. 20, Neace will lead a 12-mile walk across Perry County to raise awareness for health issues and to help himself and others become the role models he knows Southeastern Kentucky needs.
Although he doesn't consider himself a good role model, Sam Neace organized a 25-mile walk across Perry County last year and a 12-mile walk this year. He says 'People need to know they are capable of doing these things.' (Photo by Cris Ritchie)
"I have to stop and think that I am so aware of these statistics and I want to do something about it, but it is so hard for me to change me. "Therein lies a big part of the problem. And the solution," admits Neace, who as director of community engagement for local radio station WMMT 88.7 FM is a well-known figure in Perry County.
Last year, after the Today Show featured a report about the life expectancy of women across the nation. Perry County, Kentucky, ranked last. Perry County came in next to last, healthwise, in the state last year, statistics show. And for this year, Neace says they rank dead last, no pun intended.
In response, Neace quickly called the community together last fall. He organized a 25-mile walk across Perry County, garnered support for booths to be set up in Vicco, a small community outside Hazard, and offered free health screenings and help with back-to-school supplies for some of the local students. Last year, about 200 people attended the health fair. Fourteen people walked, some of those going the entire 25 miles in about seven hours.
"We all looked like we stepped straight out of war, we looked so rough," Neace remembers. "We can usually walk four miles an hour for the first three hours, then three miles an hour, and then about two miles an hour toward the end. The last five miles of last year's walk was pure willpower."
Walking 25 miles across the Appalachian foothills of Perry County on wide-shouldered highways might seem like an impossible feat to most people. But Neace is accustomed to a little challenge. He's also accustomed to being a fighter. He went head-to-head against cancer – and won.
In 2008, Neace was diagnosed with testicular cancer. He had five different types of tumors in his body. At stage III, surgery wouldn't be enough to save his life, and Neace made the decision to endure a 12-week course of three different types of chemotherapy in addition to having a testicle removed.
"I just looked at is as I've either got a 100 percent chance of living, or a 100 percent chance of dying," Neace says. "There were so many milestones. I turned 30 during the treatment, which I looked at as the start of a new decade, a new life. My wife, Rebekah, and I were building a new home, we were expecting our third child."
It was that tenacity and strength of will that helped him beat the cancer, live to love his family and learn how to help others along the way.
Those qualities also helped him to pursue his writing dreams. On Sept. 22, 2009, armed with a backpack, extra shoes and a finished manuscript, Neace set out on a 500-mile trek to Harpo Studios in Chicago, home to Oprah Winfrey's show. It was his intention to hand-deliver his manuscript and have the Queen of Talk Shows read his book.
The Great Perry County Walk includes a health fair and coincides with Hazard's Black Gold Festival. (Photo by Johnny Cummings)
When he arrived in Chicago, he learned that Oprah was out of town, but an executive took his manuscript. Neace never heard back from that endeavor, but he did learn one important thing – he could walk, rain or shine, 500 miles.
That walk started with a dream – and a single step. This Saturday's second annual Great Perry County Walk also also starts with a single step. It begins at 9 a.m. on the Perry/Knott County line on Highway 80 at Jamestown and is scheduled to end just in time for the Community Health Fair in the Perry County Park, off Highway 15 in Hazard.
So far, 20 people have signed up to walk the 12-mile stretch, but Neace says he believes more people will show up the day of the walk.
The walk and health fair are sponsored by the radio station where Neace works and is host of several programs, including Mountain Survivors, a talk show that features cancer survivors and their stories. The National Testicular Cancer Society and the Lexington Cancer Foundation are also helping out with support. The health fair booths will offer glucose screenings and blood pressure readings, as well as information from local organizations and businesses such as the Hazard Clinic, Vicco Dental and a local chiropractic practice.
"Walkers are invited to walk as much as they can, but they don't have to walk the entire 12 miles. It's not a race, it's just a statement," Neace says of the walk that has no time limits, no big media coverage, and no trophies or awards for the ones who make it across the finish line. "It's mostly just the opportunity to say they did it. Ultimately what you really want to happen is that you want someone to say they will start walking a little bit. Just keep doing things."
The walk and the health fair coincide with Hazard's Annual Black Gold Festival, a weekend event where festival goers celebrate their history and heritage with the coal industry and listen to live entertainment from popular regional musicians.
Neace says he hopes the health fair and the festival will have crossover traffic this Saturday, something which might lead to a healthier community.
"It's a culture issue here in Southeastern Kentucky. Our culture has developed some unhealthy habits. We have a high percentage of tobacco use and we are a fast food culture," Neace says. "I live in Hazard. They rank towns around here as far as how many fast food restaurants it has. We have a Burger King, a McDonald's, a Hardee's and all. Culturally speaking, we have a lot against us."
Sam Neace: 'Wouldn't it be kinda neat if the whole world was walking on the same day?' (Photo by Cris Ritchie)
Despite the challenges, Neace believes it is the small things that lead to big changes over time.
"If we can get others to start doing the little things, it will make a difference there too. People tend to look at this like it is an overwhelming task," he says. "The statistics are true, but they don't reflect this place at all. There is a whole movement going on right now toward being healthier."
Grow Appalachia, an organization funded by John Paul DeJoria, is helping community members throughout the Appalachian region learn how to plan, grow and harvest from community gardens. The program is also helping with funding and education and build monies for what are called High Tunnels, greenhouses that will allow a longer growing season, and will offer more "food security" for low income communities.
And that's just one organization; there are multiple movements in the region. The Foundation for Appalachian Kentucky gives a philanthropic push toward healthy community initiatives. A Hazard group called Pathfinders is dedicated to hiking, walking and healthy moving for all ages and has started work on a walking trail to benefit Perry County citizens.
Lott's Creek Community School, an original settlement school, has joined with Grow Appalachia to help with 20 community gardens throughout the area. The Appalachian Regional Commission's food tourism initiative, Bon Appetit Appalachia, has worked at providing an interactive map featuring nearly 50 Eastern Kentucky fresh and healthy food locations which should not only bring in tourism dollars, but is hoped to increase local awareness of good food choices.
As for Neace, he realizes his journey does, indeed, begin with a single step.
"I am a very imperfect person. If I can do anything, then anybody can. I'm just a regular guy. People need to know they are capable of doing these things if they just set their minds to it," Neace says. "As far as the future of the Great Perry County Walk, I'm kinda the person who says, however healthy it can be, however big it can be, let's make it that. Wouldn't it be kinda neat if the whole world was walking on the same day?"
Kristy Robinson Horine is a freelance journalist who lives in Paris.
Tags:Appalachian Regional Commission · Bon Appetit Appalachia · Cris Ritchie · Eastern Kentucky · Grow Appalachia · Hazard Annual Black Gold Festival · Hazard Clinic · Hazard Kentucky · Johnny Cummings · Kristy Robinson Horine · Lexington Cancer Foundation · Lott's Creek Community School · Mountain Survivors · Pathfinders · Perry County health statistics · Sam Neace · Southeastern Kentucky · testicular cancer · The Great Perry County Walk · vicco · Vicco Dental · WMMT 88.7 FM
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text:
Biografia
Era il primo figlio maschio di William Lyttelton, III barone Lyttelton, e di sua moglie, Lady Sarah Spencer, figlia di George Spencer, II conte Spencer. Studiò a Eton College e al Trinity College di Cambridge.

Carriera
Egli succedette al padre come quarto barone Lyttelton nel 1837 e prese il suo posto nella Camera dei lord al suo 21º compleanno, un anno dopo.

Nel gennaio 1846 è stato nominato Sottosegretario di Stato per la Guerra e le Colonie nel governo conservatore di Sir Robert Peel, incarico che ha ricoperto fino alla caduta del governo, nel giugno dello stesso anno. Lyttelton è stato anche Lord Luogotenente di Worcestershire (1839-1876) e il primo presidente di Birmingham e Midland Institute nel 1854. Inoltre, ha fondato la regione di Canterbury, Nuova Zelanda, con coloni anglicani. Il porto di Canterbury porta il suo nome. È stato presidente della English Chess Federation al momento della controversia Staunton-Morphy nel 1858.

Matrimonio

Primo Matrimonio
Sposò, il 25 luglio 1839, Mary Glynne, figlia di Sir Stephen Glynne, VIII Baronetto ed Mary Griffin, cognata di William Ewart Gladstone. Ebbero dodici figli:

 Lady Meriel Sarah (1840-1925) sposò John Gilbert Talbot, ebbero dieci figli;
 Lady Lucy Caroline (1841-1925) sposò Lord Frederick Cavendish, non ebbero figli;
 Charles Lyttelton, VIII visconte Cobham (1842-1922);
 Lord Albert Viktor (1844-1928);
 Lord Neville Gerald (1845-1931), sposò Katherine Sarah Wortley, ebbero tre figlie;
 Lord George William Spencer (1847-1913);
 Lady Lavinia (1849-1939) sposò Edward Stuart Talbot, ebbero cinque figli;
 Lord Arthur (1852-1903), sposò Kathleen Mary Clive, ebbero tre figli;
 Lord Robert Henry (1854-1939), sposò in prime nozze Edith Santley, sposò in seconde nozze Oliva Clarke, non ebbe figlia da entrambi i matrimoni;
 Lord Edward (1855-1942), sposò Caroline Amy West, ebbero due figli;
 Lord Alfred (1857-1913), sposò in prime nozze Octavia Tennant, ebbero un figlio, sposò in seconde nozze Edith Sophy Balfour, ebbero tre figli;
 Lady Mary Catherine Lyttelton (1857-21 marzo 1875).

Secondo Matrimonio
Sposò, il 10 giugno 1869, Sybella Harriet Clive, figlia di George Clive e Anne Farquhar. Ebbero tre figlie:

 Lady Sarah Kathleen Lyttelton (12 maggio 1870-1º ottobre 1942), sposò John Bailey, non ebbero figli;
 Lady Sybil Lyttelton (17 febbraio 1873-2 ottobre 1934), sposò Sir Lionel Cust, ebbero un figlio;
 Lady Hester Margaret Lyttelton (26 dicembre 1874-26 marzo 1958), sposò il reverendo Cyril Alington, ebbero tre figli.

Morte
Si suicidò il 19 aprile 1876, a 59 anni, gettandosi giù per le scale nella sua casa di Londra.

Onorificenze

Note

Altri progetti

Collegamenti esterni

George
George Lyttelton, IV barone Lyttelton

redpajama_set_name: RedPajamaWikipedia


From a co-creator of the Emmy-nominated web series Brown Girls comes an imaginative, soulful debut poetry that collection captures the experiences of being a young Pakistani Muslim woman in contemporary America. Orphaned as a child, Fatimah Asghar grapples with coming of age and navigating questions of sexuality and race without the guidance of a mother or father. These poems at once bear anguish, joy, vulnerability, and compassion, while also exploring the many facets of violence: how it persists within us, how it is inherited across generations, and how it manifests itself in our relationships. In experimental forms and language both lyrical and raw, Asghar seamlessly braids together marginalized people's histories with her own understanding of identity, place, and belonging.


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Home > News and Publications > JHM Publications > Dome > Dome April 2013
Dome - Getting a Lift to Safety
Dome April 2013
Getting a Lift to Safety
At Suburban Hospital, the safe use of an automated lifting device to transfer a "patient" (played by Julia April, a physical medicine aide) is demonstrated by clinical support team leader Alimany Dainkeh and nurse Mary Anderson.
Too weak to stand on her own, the patient had slunk to the floor, where she remained while a nurse sought help to lift her into a hospital bed. Happening by chance upon the scene, Quentin Simeone, clinical coordinator for patient transit services at Suburban Hospital, suggested they use the mechanical lift device that was available for that purpose.
"The patient probably weighed over 350 pounds, and there was no way we could have safely lifted her back into bed. And someone on the care team could have injured themselves," Simeone says.
In 2007, the Maryland legislature passed a law requiring hospitals to create safe patient handling policies and protocols for using mechanical devices to lift, transfer and reposition patients who weigh 35 pounds or more. The legislation chiefly targeted the high incidence of back, shoulder and other injuries experienced by nurses and other frontline staff.
But as Simeone and co-workers developed a compliance plan, they soon realized that a safe environment for employees is a safe environment for patients as well.
Whether in ambulatory or inpatient settings, unsafe handling procedures that place patients as well as workers at risk have caught the attention of caregivers throughout the Johns Hopkins Health System and across the nation. "Few activities in health care link patient and worker safety more directly than lifting, transferring, repositioning and ambulating patients," the Joint Commission asserts in a recent monograph.
The uncertainty experienced by patients during a makeshift transfer hinders trust in caregivers, says Terry Nelson, assistant director of nursing in the Department of Medicine at The Johns Hopkins Hospital. "A lot of patients look at us inquisitively, as if to ask, 'How strong are you?' and you're 5 feet, 2 inches and don't have the best back," Nelson says. "Even when patients are being moved from a stretcher to a bed or chair, they feel better when there's a device underneath them."
With guidance from Hopkins Hospital's safe patient handling committee, Nelson, along with Melinda Sawyer, then the department's safety officer, spearheaded a department-wide initiative to reduce lift-related injuries. New devices were placed on every unit, an ongoing training program launched and a cadre of nurse super-users assigned to promote unit-based safe handling practices.
"We need to change the culture: that's the most difficult thing," Nelson says. "Every single day, managers and lift champions have to make the equipment visible and advocate its use, help nurses and other frontline workers use the devices more quickly, and more efficiently."
Reinforcing the Message
This year, Hopkins Hospital's safe patient handling committee has made it a priority to educate staff on the availability of lift devices and protocols for their use. In January, the committee invited a safe patient handling expert to reinforce the correct use of lifts in various clinical settings. Slowly but surely, constant encouragement from Nelson and scores of other managers and lift champions in the Department of Medicine have led to acceptance of new handling policies and practices, she says. "As they care for a sicker and heavier population, medicine staff members on all floors are reporting regular use of mobile lifting devices," Nelson says. Correct handling methods and devices are essential in outpatient settings as well. Until recently, some frail or disabled patients arriving at the Johns Hopkins Outpatient Center didn't anticipate the level of support they would need to get out of their vehicle, even with assistance. "We would do what we could, but recognized that there has to be a safer way for the patient and staff," says Julie Kubiak, the outpatient center's assistant director of nursing.
After careful research, Kubiak's department purchased a portable lift and partnered with security personnel to develop a protocol for assessing patients' need for lift assistance. Before the lift device went into service, however, it got a workout from Kubiak and her team. "We brought our cars around and pretended to be patients to see what it would feel like, located the flattest part of the curb and how the process differed, depending on the vehicle."
Portable lifts are now stationed outside the adult emergency department in the Sheikh Zayed Tower, the Meyer building and the Jefferson Street entrance to the Weinberg building. "Some people may question the need for a lift that is used perhaps four or five times a year. But every time it's used, it's very important," Kubiak says. "Otherwise, that patient wouldn't be able to see their provider."
—Stephanie Shapiro
The $150 Million Performance-Improvement Initiative
Chief Perspectives
Building a Renaissance
Epic: Off to a Good Start
A Strong Support System
It's a Wrap
Eternally Grateful
Safe Practices
Ideas at Work
Indulgent Wellness
Around Hopkins Medicine
Striking a Match
Infection Control Triumphs
Radiothon Raises Nearly $872,000
Attention Asian and Pacific Islander Employees


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František Vorlíček (* 3. března 1947) je bývalý český hokejový útočník. Po skončení aktivní kariéry působí jako trenér.

Hokejová kariéra
V československé lize hrál za Duklu Jihlava, TJ Sparta Praha a TJ Zetor Brno. Odehrál 11 ligových sezón, nastoupil ve 347 ligových utkáních, dal 90 ligových gólů a měl 86 asistencí. S Duklou Jihlava mezi léty 1967 a 1972 získal 6 mistrovských titulů v řadě. V nižších soutěžích hrál i za TJ Kolín a I. ČLTK Praha. Za československou reprezentaci nastoupil 12. března 1969 v Pardubicích proti týmu USA.

Klubové statistiky

Trenérská kariéra
Jako hlavní trenér vedl v nejvyšší soutěži HC Kometa Brno.

Externí odkazy

 Od Rudé hvězdy ke Kometě
 Historie světových hokejových turnajů a zlínského hokeje
 Bylo to jako hrát ve fotbale za Bayern, vzpomíná Vorlíček na Jihlavu
 Dukla v extralize? To jsem se smál, říká bývalý kouč jihlavských hokejistů
 Trenér Vorlíček: Říká se v nejlepším přestat
 Trenér Vorlíček získal srbský titul, práci u tamního národního týmu ale nakonec odmítl

Českoslovenští lední hokejisté
Čeští lední hokejisté
Hráči HC Kometa Brno
Hráči HC Dukla Jihlava
Hráči HC Sparta Praha
Hráči HC Kolín
Hráči I. ČLTK Praha
Trenéři HC Dukla Jihlava
Narození 3. března
Narození v roce 1947
Žijící lidé
Muži


text:
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Fisker Automotive Inc. is an electrical vehicle automaker. They were one of the world's first plug-in hybrid electric vehicle manufacturers, which culminated in the creation of the Fisher Karma. It debuted at the 2008 North American International Auto Show. Production of the Fisher Karma began in 2009 but was suspended in 2012 with a total of 2,450 vehicles built. In 2014, Fisker was purchased by Wanxiang America Corp, a Chinese auto parts maker and the company was renamed New Fisker. Fisker is seen as the only company in position to compete with Tesla for the emerging electric automotive market. Fisker will continue production in 2016 and has selected Power Systems & Controls to help support their California plant's Frequency Conversion needs.
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The purpose of this study is to develop a conceptual framework towards building a model for venture capitalists to evaluate investments in technology based firms, considering them from their growth stage. Fast growth entrepreneurs need this model to determine the right investors to approach, since not all specialize in funding early stage high risk companies. Previous studies revealed that major problems TBF owners face is identifying financial institutions to approach for funding, and what criteria that financial institutions used to evaluate the technologies? And also, several capital are lying idle with retired and successful individuals who also encounter difficulty of identifying the right companies to fund, because they do not possess the requisite skills to evaluate businesses based on their proposals. Past research have failed to represent the growth stages of the TBFs in a model such as this, which makes it easier for interest parties to adopt when evaluating thousands of business proposals, though they all mentioned the various growth stages except the Death stage mentioned in this study. Literature review is from secondary sources such as journals, textbooks, e-journals, websites, newspaper articles, online materials, personal discussions with colleagues. The findings of this study are important because it contributes more insights to the academic research of financing technology based firms, and call for further research on presenting the growth stages using graphical approach. Furthermore, the result of this research will be useful to link theory to practice.
Technology Based Firms; Growth Cycle; Venture Capital Finance; Business Angels.


Julie Gard is a prose poet that lives in Duluth and teaches writing at the University of Wisconsin-Superior. She writes about art, travel, family, and water.
Potter/filmmaker Mark Lambert & his crew have created a documentary, "Warren MacKenzie: A Potter's Hands," about the process and work of this remarkable artist. WTIP's Buck Benson spoke recently with Mark to learn more about the film, which is being screened Friday, June 6 at 5 p.m. at the Grand Marais Art Colony.


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Recently, I read a testimony about a woman whose son was in a serious car accident and was told by the doctors that he would not survive. Overcome with grief, she prayed and asked God to spare her son's life, even if he was left severely crippled. Immediately, she was convicted by the Holy Spirit, who prompted her to pray for a complete recovery for her son. She obeyed and as a result, her son was restored to wholeness in a way that even the doctors called "miraculous." I can't help but wonder how different things would have turned out for this woman and her family if she had not been sensitive to the Spirit of God's leading. She could have easily listened to and agreed with the negative reports that surrounded her, especially by the medical experts who had examined her son. She could have caved into her fears and feelings of desperation and despair. And she could have seen the visible damage done to her son's body and concluded--as most of us would have--that he was doomed. Instead, she boldly embraced God's invitation to lay hold of a miracle for her child. I believe that, too often, we let our minds, our circumstances, or other people tell us how to pray. As a result, we often settle for less than God's best. The Bible says, "The Lord is able to give you much more than this." (2 Chronicles 25:9 NKJV) This is God's invitation to go from good to better, and from better to best in every area of our lives. Prayer is an essential part of the process and that's why it's important for us to depend on God's Spirit to help us pray. Very often, I'm prompted by the Holy Spirit to pray for the "impossible." Invariably, my own spirit cries out, "But what if God doesn't answer my prayer?" The answer that always comes is, "What if He DOES?" I've decided that I'd rather ask and risk disappointment than to never know if the Lord would have used me to lay hold of a miracle for someone. What circumstances are you facing today that could use a touch of the miraculous? Today, ask the Lord to lead you in prayer, then get ready to reap an abundance of blessings!
Prayer: Lord, teach me to ask for Your guidance when I pray for myself and others. Help me to become sensitive to Your Spirit's leading by seeking You daily in prayer, praise, and Bible study. Give me the faith and the grace I need to reach for Your highest and best in every situation. Thank You that my Spirit-led prayers will bless many and glorify You!


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Q: Time limit exceeded in C I did the code for insert the elements in the linked list its working properly.Then i wrote the code for reverse a linked list in groups of given size.My problem is my code takes too much of time for execution because i want to optimize this code.
For ex:
INPUT
8  //number of elements

1 2 2 4 5 6 7 8  //elements

4  //position to reverse the list elements

OUTPUT
4 2 2 1 8 7 6 5

struct node *reverse (struct node *head, int k)
{
    int count=0,i=0,j=0,a[1000],start=0,end;
    struct node *temp=head;
    struct node *t=head;
    struct node *tt=head;
    while(temp!=NULL)
    {
        temp=temp->next;
        count++;
    }
    while(count!=0)
    {
    while((k--)&&(t!=NULL))
    {
        a[i++]=t->data;
        t=t->next;
    }
    end=i-1;
    while(start<end)
    {
       int temp=a[start];
       a[start]=a[end];
       a[end]=temp;
       start++;
       end--;
    }
    while(tt->next!=t)
    {
        tt->data=a[j];
        j++;
        tt=tt->next;
    }
    tt->data=a[j];
    count=count-k;
    i=0;
    j=0;
    }
    return head;
}

A: The code needed is literally 6!! lines long
Here you have my example
#include <stdio.h>

void reverse(int *first, int *last)
{
    int tmp;

    while(first < last)
    {
        tmp = *first;
        *first++ = *last;
        *last-- = tmp;
    }
}

void reverseFromPos(int *arr, size_t len, size_t pos)
{
    reverse(&arr[0], &arr[pos - 1]);
    reverse(&arr[pos], &arr[len -1]);
}

int main()
{
    int arr[] = {1,2,3,4,5,6,7,8, 9};
    size_t len;

    reverseFromPos(arr, len =sizeof(arr) / sizeof(arr[0]), 4);
    for(size_t index = 0; index < len; index++)
        printf("%d ", arr[index]);
    printf("\n");

    return 0;
}

redpajama_set_name: RedPajamaStackExchange


Several weeks ago I posted Leigh Donovan's Women Involved interview. She talked of her mountain bike skills clinic-which you can learn more about at ichoosebikes.com!
"I love teaching and educating women about riding, racing, and the industry and since we need to even the scale out a bit more and if you're a single woman you can't beat the chances you'll have at the bike races/events!"
I can't be specific but I remember at 5 my dad found this awesome banana seat bike with streamers at a garage sale and I rode that thing for about 5 years everywhere.
I began racing BMX bikes when I was 11 years old (1983). I loved racing the boys, I was always more competitive with the boys than the girls (because the girls were all my friends and the boys weren't as nice when you're young). At age 20 I was introduced to Mountain bike racing and by age 23 I was World Downhill Champion and 2 time US national champion. I competed because it was the only time I wasn't scared when I was going fast on my bike.
During my glory years I loved racing the UCI World Cup events, but today I love the Crankworx event in Whistler, B.C. (Canada)...in my opinion it's the funnest and most social event of the year!!!
My favorite kind of riding is Downhill mountain biking when you're taken up the hill by chairlift and given beautiful trails to ride down. I prefer jumps, rock gardens, and technical corners but I also do love a great cross country ride in some of my favorite towns like Park City, Utah and Whistler, B.C.
I do remember my first mountain bike ride, it was on my local trails (I still ride today) and I struggled with the heat, the uphills, and trying to keep up with the riders I was with. However, the downhill was AMAZING and I blew right by those riders that smoked me up the hill and I thought to myself, if I can ride up this hill a little faster I might be pretty good at this (that was before I knew there was an official category just for downhill). I still managed to get pretty good at racing cross country too, but never quite as gifted at that as the downhill.
I have a ton of fear. I'm afraid of heights (which made riding the gondolas to the tops of the hill to race a bit scary for me), but I learned self hypnosis which really helped me overcome my weakness and actually look forward to the things that scared me most. Being nervous is good and I've learned to embrace it as a positive...sometimes it's the humble side of you that keeps you safe!
I love clipless pedals, but they take some time to get use to. I suggest for a new rider to the pedals that you ride on a grass field and practice putting your foot into the pedal and pulling out of the pedal (many times). I recently change pedals after riding the same manufacturer for 20 years and I started on a grassy area to get use to the new pedal. Clipless pedals might look intimidating but honestly if you ride on grass and practice the motion of putting your foot in and out you'll see how much they will improve your ride.
I've had so many "biff's" on my bike, but riding bikes is my therapy and I couldn't live without it. I tell riders who have had a rough crash to start back at the basics. Ride easy trails, bike paths, or anything that feels easy to you. Once you're back consistently then you can start moving forward toward the riding you were doing before. Never let a crash beat you...it may take you 3 years to get ride that section, but NEVER let it win! You shouldn't give up something you love and that empowers you just because you hurt yourself.
I could probably write a book about what I love about riding my bike, but in one easy sentence...Riding my bikes completes me.
I ride for Liv Cycling (which is the female specific bike of Giant USA). The bikes are 27.5" wheel bikes (the new wheel size is the best I've ever ridden and suggest your readers when purchasing their next bike to highly consider this new wheel size). I ride an XC bike (the Lust) and an Enduro bike (the Intrigue). The Lust is more of a race bike (very light with not a lot of suspension), while the Intrigue is for riding up hills and downhills faster, it has lots of suspension to soak up the bumps and is just a lot of fun to ride. I feel honored to work with a brand that supports women specific at all levels and in my opinion they make beautiful bikes!
I work with Troy Lee Designs and was their first rider back in 1998 to compete in their race gear. I think their gear fits well, is well made and is well priced, but I also am impressed with brands like Voler (great chamois's), GORE-tex (expensive but great fit and well made), and the Liv clothing is beautiful and fits awesome! I would recommend these 4 brands if you're in the market for a quality product.


text:
Welcome home, hero - here's your rates bill
"To all these the town owes a debt of undying gratitude." Percy Blundell's tribute to Lutonians who had fought in the Great War was published in The Luton News (April 3rd, 1919) as a preface to a poem that offered a very different light on a Corporation approach to an ex-serviceman. The poem, bearing the name Arthur R. Child and reproduced here, is entitled The Payment:
Broken in war, the lad came back,
One of many of Army's wrack,
To his home, and bed of sickness sore,
Back from the welter of ruin and gore.
But still he could hear the bursting shell,
And the crash of the aero bomb as well,
For shattered nerve and weary brain
Had broken beneath the awful strain.
He had given his best for the good of the earth,
He had fought for his home and the town of his birth,
And carried an honour not freely disposed,
But soon his reception at home was disclosed.
"Back from the war, laddie, welcome home!
We have anxiously waited for you to come;
Your rates are unpaid, and we were disposed
To destrain for them – but your house was closed.
"But now you must pay, of that there's no doubt,
Fire may have been lit – house would spoil without;
Your service to us and your country is naught,
It does not concern us, or how you have fought.
"It may be you're shattered in nerve and in health,
But you really must pay and add to our wealth;
We did not ask you to fight for us,
So pay up at once without any fuss."
So the broken lad paid, and with it a curse
On those who could not have treated him worse.
And God help our country in future's woe
If this is the payment for debt we owe.
The poet was in fact Robert Arthur Child, a former champion cyclist and skater, who ran a hat manufacturing factory in Williamson Street and lived at 37 Downs Road, Luton. And the following edition of the Tuesday Telegraph produced a response from the soldier, F.R.C. - or in other words his son Francis Robert (Frank) Child, who was convalescing from his wartime experiences at Eastbourne.
In a letter to the Tuesday Telegraph, Frank wrote: "In The Luton News forwarded to me here (where I am staying in hope of regaining my broken health due to war), I notice that Mr Percy Blundell paid a splendid tribute to the men who fought and suffered for home and country. He concludes: 'To all these the town owed a debt of undying gratitude'.
"Mark the ' gratitude' shown by the local authorities to a townsman broken in war. I volunteered for service and was A1 category, served 2½ years in Belgium and France, including eight months in the 'hell' at Ypres. When I joined the Army, my house was closed up, and my wife (no children) was compelled to live with friends, as she could not live alone.
"I left good income, home and business to serve for 2s 4d per day, and recently returned nerve shattered, utterly broken up and unable to work or resume business.
"Immediately the overseers knew I had returned they demanded dull accumulated rates, which I refused to pay, on ground that occupation of house was impossible while I was serving my country (which, I may point out to authorities, includes Luton), and that nothing whatever had been done to justify demand, especially as Downs is a private road.
"They summoned me, and being certified by the doctor too ill to appear, I was represented (by the kindness of my father) by a solicitor. During the hearing of the case the brilliant idea that fires had been lit in the house to prevent spoilation (which I am sorry to find did occur to some extent), occurred to one of the magistrates, and on this assumption, and the decision that this constituted occupation, an order was made for full amount of rates (and costs) accumulated on an unoccupied house while owner was helping to defend his and their homes. An order was also made for immediate distraint in case of non-payment.
"I have paid, so the full gratuity given by the nation for 2½ years service, and addition to make up the amount, goes to help swell the £22,000 the Corporation have accumulated, and for which they are indebted to the men who have sacrificed themselves to save the country (and Luton) from the Hun.
"Would that they had had a taste of the 'hell' we have been in, and seen the trail of the Hun in Belgium and France. There is nothing to rate there, and, but for us, it would have been so in England (or worse). We should have been under the German heel and lost all worth living for.
"God help England in the future if all towns were to treat men broken in war as Luton is doing."
Frank Child, born on February 15th, 1888, recovered from his experiences and rejoined wife Lillie, whom he had married at St Pancras Parish Church on July 14th, 1914, and they had a son, Geoffrey. Frank had been an assistant in his father's hat business and took over the business on his father's death in March 1939. He died on November 13th, 1957, while residing at Sharpenhoe.


Following Tiger Woods' epic win at The Masters on Sunday, some of the biggest sports stars in the world took to Twitter to celebrate and congratulate the golf great. As Sports Illustrated reports, Tom Brady commented on Woods' 15th career major by tweeting, "Running the numbers on how long it'll take me to get to 15…Congrats Tiger! What a performance." Meanwhile, Serena Williams wrote, "I am literally in tears watching @tigerwoods. This is Greatness like no other. Knowing all you have been through physically to come back and do what you just did today? Wow. Congrats a million times! I am so inspired. Thank you buddy." And Steph Curry wrote, "Greatest comeback story in sports!" As The New York Times reports, Woods even managed to get the bipartisan Twitter congratulations of both of the most recent golf-loving Presidents, Barack Obama and Donald Trump.


text:
Fundamental analysis: AirSculpt Technologies, Inc. (AIRS)
The higher the Awarener score, the more bang you get for the buck. It measures how much genuine funds the company generates for the stock price paid (could not be estimated), the business stability (unknown) and growth (unknown), and the company's inclination to return cash to the stockholders (unknown).
AirSculpt Technologies, Inc. business stability could not be estimated, due to insufficient input data. It looks we cannot compare it to rivals.
AIRS profit margins -on goods and services sold- are usually excellent. They stand well ranked against rival companies.
Business profit on sales tends to be meagre. It's substantially worse when measured against competitors.
Profits on sales made -available to repay debt and purchase properties- are usually meagre. They remain in a weak position compared to peers.
Earnings -before income taxes and interests on loans taken- tend to be meagre in relation to total revenues. They're still worse than most similar companies.
Profits -before income taxes- are usually meagre considering total sales, and remain weak when measured against rivals.
Total net profit tends to be meagre when confronted to sales. Company stands weak when measured against comparable firms.
AirSculpt Technologies, Inc. has an unknown gross margin growth, as there is not enough data to analyze. It's been impossible to compare to competitors.
AIRS had still to pay income taxes, even though in recent past years mostly lost money. It's been bottom tier against peers.
AirSculpt Technologies, Inc. usually gets pauper returns on the resources it controls. It proves last-in-rank when measured against peer firms.
The company normally gets extremely poor proceeds -on the resources directly invested in the business-. They remain a disappointment compared to similar companies.
There's usually bottom profitability -in relation to owned resources-. It ranks last-in-rank when measured against competitors.
In the past, got pauper returns -on the tangible resources it controls-. This metric is usually related to the industry in which operates and combines profitability versus reinvestment needs. It's last-in-rank when measured against comparable enterprises.
AIRS remains pending of analysis regarding capital expenditures, due to data unavailable. It stands last-in-rank when measured against rival firms.
The company is usually investing in new property, plant, and equipment, to improve its operating capabilities, which is more than average in relation to industry peers.
In the past twelve months it paid very little dividends, considering the current stock price. It came mediocre against competitors.
Has recently started or restarted paying dividends to stockholders. Business prospects are most likely good. The company has behaved impressive in relation to similar firms.
Shares count analysis is unavailable, on lacking data. It remains we cannot compare it with peer enterprises.
We are not sure on the effectiveness of the company when repurchasing shares, as there were not enough numbers to crunch. It stands unidentified against rivals.
AirSculpt Technologies, Inc. intangible assets (like brands and goodwill) represent a huge portion of resources controlled, according to accounting books. There could be major difficulties in liquidating them if the company ever gets in financial distress. It happens to be weak when measured against peer companies.
The company has lower short-term resources than short-term obligations. Unless it's part of the business model, there might be liquidity concerns. It turns to be a disappointment compared to similar firms.
A substantial part of resources controlled were provided for with financial debt. Creditors have as many claims on the company as shareholders. The situation is somewhat risky. It remains mediocre against rival firms.
Controlled resources might be only very slowly turned into cash and equivalents, which is riskier. It looks last-in-rank when measured against rivals.
For every dollar of short-term obligations, the company has less than a dollar of cash and short-term receivables. It's a disappointment compared to peer firms.
For every dollar of short-term obligations, the company has almost another of cash and equivalents, which is slightly better than similar enterprises.
Usually, sales are mostly on cash. It still ranks top tier when measured against peers.
Normally has no inventories. It comes up as impressive in relation to competitors.
On average, it takes less than one month from the purchase to charging customers. It happens to be top-notch against peers.
On average pays suppliers during the first couple of weeks from the purchase. It ranks last-in-rank when measured against industry peers.
The company pays its suppliers almost when charging its customers, so there's very little money invested in working capital. It's a slight improvement compared to similar companies.
Has usually been losing money on the business, so net interest expenses must be paid by increasing borrowings, which is unsustainable in the long run. The situation is very risky for both creditors and shareholders, profitability must increase. It stands bottom tier against rival firms.
Resource exploitation is huge considering yearly sales, which is great. This metric is normally tied to the industry where the firm belongs. It's still top-notch against peer companies.
AirSculpt Technologies, Inc. reported losses, so valuating it in relation to earnings is meaningless. It happens to be last-in-rank when measured against competitors.
Price-to-Tangible-Book-Value is a fairly complex metric. Run again in analytical mode if you're interested in a technical explanation. It remains a disappointment compared to peers.
There is insufficient information on the genuine funds generation capability showed in the past twelve months, which stands as an incognita in relation to similar companies.
Unfortunately, lack of enough yearly data impaired our ability to estimate the normal earnings power. It's still an unknown variable to measure against industry firms.
A conclusion regarding usual company rewards to stockholders was impossible to reach with data available. It came up also not enough to relate to peer ventures.
The company has neither net debt nor net cash. It may borrow extra money if it wishes so, or start cumulating cash for future uses. It looks better than most similar enterprises.
Comparing the current stock price with the past twelve-months revenues gives a three or four to one relationship. This is an important metric to check its evolution through time, and to compare to industry peers. It looks in a very weak position compared to rival firms.
The relation between the stock price and accounting book value is extremely high, which may be good or bad depending on context. Run again in analytic mode if you want to dig deeper. The company remains worse than most peer firms.
In the past twelve months, the operating business lost some money. It happens to be weak when measured against industry peers.
In an alternate metric of bang for the buck, the company has usually shown a somewhat low earnings power ability when measured against the current stock price and financial position. It's still in a weak position compared to peer companies.
Company at a glance: AirSculpt Technologies, Inc. (AIRS)
Sector, industry: Healthcare, Medical Care Facilities
AirSculpt Technologies, Inc., together with its subsidiaries, focuses on operating as a holding company for EBS Intermediate Parent LLC that provides body contouring procedure services in the United States. It offers custom body contouring using its AirSculpt procedure that removes unwanted fat in a minimally invasive procedure. The company provides fat removal procedures across treatment areas; and fat transfer procedures that use the patient's own fat cells to enhance the breasts, buttocks, hips, or other areas. Its body contouring procedures also include the Power BBL, a Brazilian butt lift procedure; the Up a Cup, a breast enhancement procedure; and the Hip Flip, an hourglass contouring procedure. As of March 10, 2022, it operated 19 centers across 15 states. AirSculpt Technologies, Inc. was founded in 2012 and is headquartered in Miami Beach, Florida.