text:
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\section{Introduction}

A \defn{polytope} is the convex hull of a finite set of points in a real Euclidean space. Its \defn{combinatorial type} is given by its poset of \defn{faces} (subsets of the polytope maximized by linear functionals, ordered by inclusion).
In the preface of his now classical book in polytope theory~\cite{G03}, Gr{\"u}nbaum
traces the problem of enumerating the number of combinatorial types of polytopes back to Euler, and cites its difficulty as one of the main reasons for the ``\emph{decline in the interest in convex polytopes}'' at the beginning of the XXth century.

These efforts were concentrated in the case of $3$-dimensional polytopes, starting with many contributions by Cayley and Kirkman, according to Gr{\"u}nbaum's historical remarks in~\cite[Chapter~13.6]{G03}. Thanks to Steinitz's Theorem, which gives a correspondence between combinatorial types of $3$-dimensional polytopes and $3$-connected planar graphs, nowadays we have quite precise knowledge on the number of $3$-polytopes with $n$ vertices~\cite{BenderWormald1988,RichmondWormald1982} and the distribution of many combinatorial parameters~\cite{BenderGaoRichmond1992}.

In contrast, for higher-dimensional polytopes the problem it is still very far from being solved. One of the main difficulties lies in the lack of a combinatorial characterization of face lattices of polytopes. Mn\"ev's Universality Theorem~\cite{Mnev1988} and its extension by Richter-Gebert~\cite{RichterGebert1996}, imply that deciding whether a poset is the face lattice of a $4$-dimensional polytope is computationally hard ($\exists\mathbb{R}$-complete). It seems thus that a simple combinatorial characterization is impossible, which is one of the intrinsic difficulties of the enumeration problem. The problem remains hard even when restricting to the ``generic'' case of \defn{simplicial} polytopes, where all faces except the whole polytope are simplices (equivalently, polytopes whose combinatorial type does not change when the vertices are perturbed), see~\cite{AdiprasitoPadrol2017}.

However, the mere number of polytopes is  relatively small.
In 1986 Goodman and Pollack~\cite{GP86} showed that the number of (labeled) combinatorially different simplicial $d$-polytopes with $n$ vertices is bounded by $(n!)^{c_d}$ for some constant $c_d$ depending solely on $d$, and Alon~\cite{Alon1986} proved that this upper bound is valid for non-necessarily simplicial polytopes too. This contrasts with the number of combinatorially different simplicial $(d-1)$-spheres with $n$ vertices, which grows at least as $e^{\Omega(n^{\floor{d/2}})}$~\cite{Kalai88,NSS16}.

In 1982 Shemer~\cite{Shemer82} had devised constructions producing about $ (n!)^{\frac12 \pm o(1)}$ different simplicial polytopes. This matches the upper bound, except for the fact that the constant $c_d$ in the upper bound of Goodman and Pollack and Alon is ${ d^2\pm o(1)}$, much bigger than the $1/2$ obtained by Shemer.
The construction was greatly improved by Padrol~\cite{Pad13} (see also~\cite{GP16}) who showed that there are at least $(n!)^{\floor{d/2}\pm o(1)}$ (labeled) neighborly polytopes. There are alternative constructions that give these many different combinatorial types of polytopes, which led Nevo and Padrol to ask whether the number of $d$-dimensional polytopes with $n$ vertices and $m$ facets was bounded above by $m^{n+o(n)}$ (unpublished). As the maximal number of facets of a $d$-polytopes with $n$ vertices is $O(n^{\floor{d/2}})$ by the Upper Bound Theorem~\cite{McM70}, this would imply that the bound of $(n!)^{\floor{d/2}\pm o(1)}$ is asymptotically tight.

The main result in this paper gives a negative answer to this question, by essentially doubling the exponent of  $n!$ in the construction of Padrol:

\begin{theorem}\label{thm:manypolytopes}
The number of different labeled combinatorial types of $d$-polytopes with $n$ vertices for fixed $d>3$ and $n$ growing to infinity is at least $(n!)^{d-2 \pm o(1)}$.
\end{theorem}

Enumerating polytopes is intimately tied to enumerating \defn{regular} triangulations of point configurations; that is, triangulations arising as lower envelopes of polytopes of one  dimension more.
In fact, the number of (combinatorial types of) simplicial $d$-polytopes with $n$ vertices coincides with that of $(d-1)$-dimensional regular triangulations with $n-1$ vertices. See the beginning of Section~\ref{sec:manypolytopes} for details on this relation.
In the same vein, counting all triangulations, regular or not, is related to counting simplicial spheres.

In particular, the Goodman-Pollack bound implies the same upper bound of $(n!)^{d^2\pm o(1)}$ for the number of  regular triangulations, while the construction of Kalai~\cite{Kalai88} can be adapted to derive that the cyclic $d$-polytope with $n$ vertices has at least~$e^{\Omega(n^{\floor {d/2}})}$ triangulations in total~\cite[Theorem 6.1.2]{DRS10}.

Observe that  the upper bound is for the total number of (combinatorially different) regular triangulations of \emph{all} polytopes (for fixed parameters $n$ and $d$), while the construction of Kalai counts triangulations of \emph{a single polytope}.
For regular triangulations of a single polytope, it is shown in \cite[Theorem 7.2.10]{DRS10} that the Cartesian product of a cyclic $3$-polytope with $n$ vertices and a segment has at least $(n/2)! = (n!)^{1/2 \pm o(1)}$ regular triangulations.  The second result in this paper is a significant improvement of this lower bound, showing  for example that:

\begin{theorem}\label{thm:manytriangulations}
For fixed $d\ge3$ and $n$ going to infinity, there are realizations of the cyclic $d$-polytope with $n$ vertices having at least
\[
(n!)^{  \ffloor{d-1}{2}  \pm o(1)}
\]
regular triangulations.
\end{theorem}

It has to be noted that the total number of triangulations of a polytope (or point configuration) depends only on its \defn{oriented matroid} (another combinatorial invariant that is finer than the combinatorial type), while the number of regular triangulations varies for different realizations of the same oriented matroid.

Apart of its intrinsic interest,
Theorem~\ref{thm:manytriangulations} is an intermediate step for Theorem~\ref{thm:manypolytopes}; the proof of Theorem~\ref{thm:manypolytopes} consists in showing that \emph{all} of the many polytopes constructed by Padrol~\cite{Pad13} admit realizations with the many regular triangulations stated in Theorem~\ref{thm:manytriangulations}.

This makes our proof  of Theorem~\ref{thm:manypolytopes}  more geometric, as opposed to combinatorial, than previous constructions of ``many'' polytopes. In fact, the combinatorial types of polytopes obtained with our method may depend on choices made along the construction, for example via the particular realizations used for the Padrol polytopes, which affects what triangulations of them are regular, or via the particular lifting vectors used for the regular triangulations.

\section{Definitions and notation}
We will follow~\cite{Zie95} and \cite{DRS10} for the terminology concerning convex polytopes, point configurations, and triangulations, and we refer the reader to these references for background on these topics.

A \defn{point configuration} is an ordered sequence $\pc =(\p _1, \ldots, \p _n)\in \mathbb{R}^{d\times n}$.
We formally consider $P$ a sequence rather than a set since the ordering of the points $\p_i$ is sometimes important,
but we will slightly abuse notation and write things like $\p_i\in P$, or call the points $\p_i$ the elements of $P$.
In this paper we will usually assume that the points in $P$ are distinct
and in \defn{convex position}, where the latter means that all of them are vertices of the polytope $\operatorname{conv}(\pc)$.
We say that $\pc$ is \defn{$k$-neighborly} if any subset of $k$ points is the vertex set of a face of~$\operatorname{conv}(\pc)$, and just \defn{neighborly} if it is $\ffloor{d}{2}$-neighborly; the latter makes sense since the simplex is the only $d$-polytope that is more than $\ffloor{d}{2}$-neighborly.

A \defn{triangulation} $\tr$ of $\pc$ is a simplicial complex on the set $[n]$ such that
\begin{enumerate}[(i)]
\item $\bigcup_{F\in \tr} \operatorname{conv}(\set{\p_i}{i\in F}) = \operatorname{conv}(\pc)$,
\item for all $F, F'\in T$, $\operatorname{conv}(\set{\p_i}{i\in F})\cap\operatorname{conv}(\set{\p_i}{i\in F'})$ is a common face of $\operatorname{conv}(\set{\p_i}{i\in F})$ and $\operatorname{conv}(\set{\p_i}{i\in F'})$.
\end{enumerate}
A triangulation $\tr$ of $\pc$ is \defn{regular} if there is a lifting vector $\p[w] \in \mathbb{R}^{[n]}$ such that
for any $F\in \tr$, $\operatorname{conv}(\set{(\p_i, \p[w](i))}{i\in F})$ is a lower face of $\operatorname{conv}(\set{(\p_i, \p[w](i))}{i\in [n]})$. Here, we call a face $F$ of a polytope~$P$ in $\mathbb{R}^{d+1}$ \defn{lower} if its outer normal cone contains a vector with last coordinate negative. That is, if there is a functional $\p[c]\in \mathbb{R}^{d+1}$ with $c_{d+1}<0$ that is maximized on $F$.

We denote \defn{$\regtriang{\pc}$} the set of regular triangulations of $\pc$.
For a triangulation $\tr$, we call \defn{cells} its maximal faces.
(These are sometimes called facets, but we reserve the word \defn{facet} for facets of a polytope).

A point $\p[q]\notin \pc$ is said to be in \defn{general position} with respect to $\pc$ if no hyperplane spanned by points of $P$ contains $\p[q]$, and in \defn{very general position} with respect to $\pc$ if moreover no small perturbation of $\p[q]$ changes $\regtriang{\pc\cup\{\p[q]\}}$.
An argument similar to that in \cite[Part 2]{Ath99} shows that configurations in very general position form a dense open subset of the space of all point configurations.

Two points $\p_i, \p_j\in \pc$ are said to be \defn{triangulation-inseparable} in $\pc$ if we have that
\begin{itemize}
\item[(i)] $\regtriang{\pc\setminus\{\p_i\}} = \regtriang{\pc\setminus\{\p_j\}}$ up to relabeling $j$ to $i$, and
\item[(ii)]  for any $\tr\in \regtriang{\pc\setminus\{\p_i\}}$ there is a lifting vector $\p[w]\in \mathbb{R}^{[n]}$ which restricted to both $\pc\setminus\{\p_i\}$ and $\pc\setminus\{\p_j\}$ produces $\tr$ as a regular triangulation (up to relabeling $j$ to $i$).
\end{itemize}

Let $\p$ be a vertex of $\operatorname{conv}(\pc)$. We define \defn{$\pc/\p$} to be any point configuration obtained as the intersection of the half-lines positively spanned by $\set{\p'-\p}{\p'\in \pc\setminus\{\p\}}$ with an affine hyperplane that does not contain $\p$ and intersects all these half-lines. Following~\cite[Definition 4.2.9]{DRS10} we call $\pc/\p$ the \defn{contraction} of $\pc$ at the point $\p$.
All the configurations that can be obtained as $\pc/\p$ have the same triangulations and the same regular triangulations.
In fact, regular triangulations of $\pc/\p$ are exactly the links at $\p$ of regular triangulations of $\pc$~\cite[Lemmas 4.2.20 and 4.2.22]{DRS10}.
Here, the \defn{link} of a triangulation $\tr$ at a point~$\p_i$, which we denote \defn{$\tr/\p_i$}, is defined as
\[
\tr/\p_i:=\set{F\subset [n]\setminus \{i\}}{F\cup \{i\} \in \tr}.
\]

\section{Many regular triangulations}

The main idea of our construction of configurations with a large number of regular triangulations is to split a point into two triangulation-inseparable points and to estimate the number of regular triangulations generated after this operation. This is inspired by the study of triangulations of cyclic polytopes done in~\cite{Rambau1997,RambauSantos2000}.

First, we show that we can indeed obtain triangulation-inseparable pairs by such a splitting.

\begin{lemma}\label{lem:triang_insep}
Let $\pc$ be a point configuration in $\mathbb{R}^d$ and $\p\in \pc$ in very general position with respect to $\pc\setminus\{\p\}$. Then there is an $\varepsilon >0$ such that $\p$ and $\p'$ are triangulation-inseparable in $\pc\cup\{\p'\}$ for any $\p'\in \ball{\p}{\varepsilon}$ in very general position with respect to $P$.
Here $\ball{\p}{\varepsilon}$ denotes the ball of radius $\varepsilon$ centered at $\p$.
\end{lemma}

\begin{proof}
Up to relabeling, we can assume that $\pc=(\p_1, \ldots, \p_n)$ and $\p=\p_n$.
By definition of being in very general position with respect to $\pc\setminus\{\p_n\}$, there exists some $\eta>0$ such that $\regtriang{\pc}=\regtriang{\pc\setminus\{\p_n\}\cup\{\p'\}}$ for all $\p'\in \ball{\p_n}{\eta}$. In particular, any such a $\p'$ fulfills the first condition for being triangulation-inseparable with $\p$.

For each regular triangulation $\tr\in\regtriang{\pc}$ we can choose a specific lifting vector $\p[w]_{\tr}\in \mathbb{R}^{[n]}$ that induces~$\tr$, and choose it so that the point $(\p_n, \p[w]_{\tr}(n))$ is still in general position with respect to the lifted  configuration $\set{(\p_i, \p[w]_{\tr}(i))}{i\in [n-1]}$. Hence there is some $0<\varepsilon_T< \eta$ such that $\set{(\p_i, \p[w]_{\tr}(i))}{i\in [n]}$ and $\set{(\p_i, \p[w]_{\tr}(i))}{i\in [n-1]}\cup\{(\p', \p[w]_{\tr}(n))\}$ have the same faces, for all $\p'\in \ball{\p_n, \varepsilon_T}$. This means that $\p[w]_T$ induces $T$ as a regular triangulation of $\pc\setminus\{p_n\}\cup\{\p'\}$ for all $\p'\in \ball{\p_n}{\varepsilon_{\tr}}$.

If we take $\varepsilon =\displaystyle{\min_{\tr\in \regtriang{\pc}} \varepsilon_T}$, we obtain that $\p$ and $\p'$ are triangulation-inseparable in $\pc\cup\{\p'\}$ for all~$\p'\in \ball{\p}{\varepsilon}$.
\end{proof}

The following result is our main technical lemma, which provides lower bounds for the number of triangulations under the presence of triangulation-inseparable points. The main ideas are illustrated in Example~\ref{ex:movingtriangulation}.

\begin{lemma}\label{lem:inductionstep}
Let $\pc$ be a point configuration  in $\mathbb{R}^d$ and let $\p\in \pc$ be a vertex in very general position with respect to $\pc\setminus\{\p\}$.
We denote $C$ the minimum number of cells in a regular triangulation of $P/p$.

Let $\p'$ be such that $\p$ and $\p'$ are triangulation-inseparable in $\pc\cup\{\p'\}$ and $\p'$ is in very general position with respect to $\pc$. Then we have
\[
|\regtriang{\pc\cup\{\p'\}}| \geq |\regtriang{\pc}| \times (C+1).
\]
\end{lemma}

\begin{proof}

We denote $\pc'$ the point configuration $\pc\setminus\{\p\}\cup\{\p'\}$.

Let us call a regular triangulation $\tilde{\tr}$ of $\pc\cup \pc'$ \emph{good} if there is a regular triangulation $T$ of $\pc$ such that $T$ and $\tilde{\tr}$ coincide when restricted to $\pc\setminus\{\p\}$.
Since a regular triangulation of $\pc$ is determined by its restriction to $\pc\setminus \{\p\}$, this definition implicitly gives a map
\[
\phi: \{\text{good triangulations of $\pc\cup \pc'$}\} \to \regtriang{\pc}.
\]
We claim that for every $T\in \regtriang{P}$ we have
\[
|\phi^{-1}(\tr)|  \ge \size{\tr/\p}+1 \ge C+1,
\]
where we denote by $\size{L}$ (and call size of $L$) the number of cells of a pure polyhedral complex $L$.
This formula implies the statement.

Let $\tr$ be a regular triangulation of $\pc$. To avoid confusion we denote by $\tr'$ the triangulation $\tr$ but considered as a triangulation of $\pc'$.
Let $w\in \mathbb{R}^{\pc\cup \pc'}$ be a lifting vector producing $\tr$ and $\tr'$ when restricted to $\pc$ and $\pc'$, which exists because $\p$ and $\p'$ are triangulation-inseparable. We will assume moreover a genericity condition on $w$.

For each $t\in\mathbb{R}$ we consider the following lifting vector $w_t\in\mathbb{R}^{P\cup P'}$, which varies continuously with $t$:
\begin{itemize}
\item For $q\in P\setminus\{p\}$, $w_t(q) :=w(q)$ is independent of $t$.
\item If $t\le 0$ then $w_t(p):=w(p)$ and $w_t(p'):=w(p')-t$.
\item If $t\ge 0$ then $w_t(p):=w(p)+t$ and $w_t(p'):=w(p')$.
\end{itemize}

Let $T_t$ be the regular subdivision of $P\cup P'$ produced by $w_t$. We have that:

\begin{enumerate}
\item \emph{The restriction of $T_t$ to $P\setminus \{p\}$ coincides with the restriction of $T$:} Indeed, if $\sigma$ is a face of $T$ contained in $P\setminus \{p\}$ then $w$, and hence any $w_t$, sends all of $P\cup P'\setminus \sigma$ above some supporting hyperplane of the lift of $\sigma$; hence, $\sigma$ is a face in $T_t$. Conversely, suppose $\sigma$ is a cell in $T_t$ for some~$t$. If $t\le 0$ then $w_t$, and hence $w$, sends $P\setminus \sigma$ above the hyperplane. Hence, $\sigma$ is a cell of $T$. If~$t\ge 0$ then $w_t$, and hence $w$, sends $P'\setminus \sigma$ above the hyperplane. Hence, $\sigma$ is a cell of $T'$, which restricted to $P\setminus \{p\}$ coincides with $T$.

\item \emph{If $t\le 0$ then for every cell $\sigma\in T_t$,  $\sigma\setminus \{p'\}$ is a face in $T$:} This is because for $t\le 0$ we have that~$w_t$ restricted to $P$ equals $w$, which produces $T$ as a regular triangulation of $P$.

\item \emph{If $t\ge 0$ then for every cell $\sigma\in T_t$,  $\sigma\setminus \{p\}$ is a face in $T$:} Same proof.

\item \emph{If $T_t$ is not a triangulation for a certain $t$ then every non-simplicial cell is of the form $\tau\cup\{p,p'\}$ where $\tau$ is a cell of $T/p$.}
Let $\sigma$ be a non-simplicial cell of $T_t$. By claims (2) and (3), $\sigma$ uses both of $p$ and $p'$ and either $\sigma\setminus\{p'\}$ is in $T$
or $\sigma\setminus\{p\}$ is in $T'$. Hence, $\sigma\setminus\{p,p'\}$ is in $T/p=T'/p'$.

\item \emph{Each such $\tau\cup\{p,p'\}$ appears as a cell for at most one value of $t$:}

If $t\le 0$ then the value of $t$ is fixed by the fact that $w_t(p')=w(p')-t$ equals the height at which the lifted hyperplane containing $\tau\cup\{p\}$ meets the vertical line $\{p'\}\times \mathbb{R}$ ; same, changing $p$ and $p'$, if $t\ge 0$.
Thus, we  at most have  one value of $t$ in $(-\infty,0]$ and one in $[0,\infty)$.
Moreover, there cannot be two values, one negative and one positive. Indeed, if $\tau\cup\{p,p'\}$ is a cell of $T_t$ for $t<0$, then the point $(p', w(p'))$ is below the lifted hyperplane containing $\tau\cup\{p\}$, so $(p, w(p))$ is above the lifted hyperplane containing $\tau\cup\{p'\}$ and there is no $t'>0$ such that $\tau\cup\{p,p'\}$ is a cell of $T_{t'}$.

\item \emph{Assuming $T_t$ is a triangulation, let $L_t:= T_t/p \setminus p'$ and $L'_t:= T_t/p' \setminus p$. $L_t$ and $L'_t$ are contained in $T/p$ and they are complementary in the sense that their union equals $T/p$ and their intersection is lower dimensional.}
$L_t$ and $L'_t$ are contained in $T/p=T'/p'$ by properties  (2) and (3). They are complementary because every cell $\tau\in T/p$ needs to be joined to one and only one of $p$ and $p'$ to give a cell of $T_t$.

\item \emph{In the limit when $t\to -\infty$ we have that $L_t=T/p$ (and hence $L'_t$ is lower-dimensional) and in the limit $t\to +\infty$ we have that  $L'_t=T/p$ (and hence $L_t$ is lower-dimensional).}
In these limits, $T_t$ equals the triangulation obtained by \emph{placing} point $p'$ (respectively $p$) in $T$ (respectively in $T'$). This implies  $L_t=T/p$ (respectively $L'_t=T/p$).

\item \emph{We can take $w$ sufficiently generic so that no $T_t$ contains two different non-simplicial cells.}
Suppose that $(\tau_1, \tau_2)$ is a pair of cells in $T/p$ such that $\tau_1\cup\{p,p'\}$ and $\tau_2\cup\{p,p'\}$ are in $T_t$ for the same value of $t$.
Let $H_1$ and $H_2$ be the two hyperplanes in $\mathbb{R}^{d+1}$ spanned by the lifts of $\tau_1\cup\{p\}$ and $\tau_2\cup \{p\}$ for that $t$. Our hypothesis implies that $H_1$ and $H_2$ intersect the vertical line $\{p'\}\times \mathbb{R}$ at the same height (namely, at height $w_t(p')$). If this happens for a sufficiently generic choice of $w$ then the intersection of $H_1$ with $\{p'\}\times \mathbb{R}$ does not change when slightly perturbing the heights of all points in $\tau_1\setminus \tau_2$: This implies that this intersection point lies in the affine span of (the lifted) configuration $(\tau_1\cap \tau_2)\cup \{p\}$. Hence, $p'$ lies in the affine span of (the original) $(\tau_1\cap \tau_2)\cup \{p\}$, and $p'$ is not in general position.
\end{enumerate}

Claim (1) says that whenever $T_t$ is a triangulation it is a
good triangulation and it lies in the preimage of $T$. As we move $t$ continuously from $-\infty$ to $+\infty$ there are finitely many values of $t$ where $T_t$ is not a triangulation, by claims (4) and (5). Of course, outside those values the triangulation $T_t$ is constant, and claim (8)  says that (if $p'$ is in general position and $w$ is generic) at those values the change in the triangulation is a geometric bistellar flip in a cell of the form $\tau\cup\{p,p'\}$. This flip changes the numbers of cells of $T/p$ contained in $L_t$ and in $L'_t$ by one unit, increasing $L'_t$ and decreasing $L_t$ as $t$ increases.
By property (7) the size of $L'_t$ grows from zero to $\size{\tr/\p}$ as $t$ goes from $-\infty$ to $+\infty$, so we encounter
at least $\size{\tr/\p}+1$ different good triangulations in the preimage of $\tr$ along the process.
\end{proof}

\begin{figure}[htpb]
        \centering
\input{Figures/proof_inductionstep}
\caption{A two-dimensional illustration of the proof of Lemma~\ref{lem:inductionstep}. In the left picture, a triangulation of a hexagon. In the rest, the vertex $p$ is split into $\{p,p'\}$, and increasing values of the parameter~$t$ induce different good triangulations of the heptagon $P\cup P'$.
}\label{fig:proof_inductionstep}
\end{figure}

\begin{example}\label{ex:movingtriangulation}
 To illustrate Lemma~\ref{lem:inductionstep}, we use a two-dimensional example (reminiscent of some classical proofs of the recurrence relation for Catalan numbers). It has the advantage of clarity, as it can be easily depicted, see Figure~\ref{fig:proof_inductionstep}.

 When a point $p$ is split into $\{p,p'\}$, the facets of the polytope that were incident to~$p$ are divided into two families, those that remain facets after the splitting, and those that replace $p$ by $p'$. Moreover, new facets containing both $p$ and $p'$ are created. Similarly, the cells incident to $p$ in a triangulation are divided into two families, those containing $p$ and those containing  $p'$, and new cells containing  both $p$ and $p'$ are created. This can be read in the link $T/p$, which is divided into two parts, $L_t$ and $L'_t$, without full dimensional intersection: We have that $F\cup \{p\}\in T_t$ whenever $F\in L_t$, $F\cup \{p'\}\in T_t$ whenever $F\in L'_t$, and that $F\cup \{p,p'\}\in T_t$ whenever $F\in L_t\cap L'_t$.

 When $t=-\infty$, we have that $L_t\cap L'_t=L'_t$, which coincides with the faces of the convex hull incident to~$p'$. As the values of $t$ increase,  $L_t\cap L'_t$ flips successively through each of the simplices of $T/p$, giving rise to different triangulations. At the end, when $t=\infty$, we have that $L_t\cap L'_t=L_t$, which coincides with the faces of the convex hull incident to $p$. The number of different triangulations thus created is therefore one more than the number of simplices in the link.

There are some important differences that only appear in higher dimensions. First of all, all triangulations of a polygon are regular, while starting in dimension 3 there are polytopes with non-regular triangulations. Moreover, in dimension two there is only one way to go from $T_{-\infty}$ to $T_{\infty}$ since the link $T/p$ is one-dimensional, while in higher dimensions there are usually several different paths between these two triangulations. The $\size{\tr/\p}+1$ triangulations $T_t$ that we see as $t$ ranges from $-\infty$ to $\infty$ will depend on the relative position of $p$ and $p'$ and the choice of the lifting vector~$w$. Finally, in a polygon, the vertex figure is just a segment (with interior points) and the number $C$ in the statement is always $1$, so the lemma does not give an interesting bound in that case.
\end{example}

Without any further constraint this lemma is not very useful, as $\operatorname{conv}(\pc/\p)$ could be a simplex and $C=1$. However, a lower bound on $C$  can be proved if we have knowledge on the neighborliness of $\pc/\p$, thanks to the following lemma.

Recall that for a pure $d$-dimensional simplicial complex $\mathcal{C}$ and $0\leq j \leq d+1$ we denote \[h_j(\mathcal{C})=\sum_{k=0}^j (-1)^{j-k} \binom{d+1-k}{d+1-j} f_{k-1}(\mathcal{C}),\] where $f_k(\mathcal{C})$ is the number of faces of $\mathcal{C}$ of dimension $k$. The numbers $h_0(\mathcal{C}), \dots, h_{d+1}(\mathcal{C})$, collectively called the \defn{$h$-vector} of $\mathcal{C}$, are known to be nonnegative in certain special cases, which include $\mathcal{C}$ being a topological sphere; see~\cite[Chapter 8]{Zie95}.

\begin{lemma}\label{lem:cells_triang_neighb}
Let $d>2$ and $1\leq k \leq d+1$.
Let $Q$ be a $d$-dimensional simplicial polytope on $n$ vertices.
Then the number of cells in any triangulation of $Q$ is bounded from below by $h_k(\partial Q)$.

In particular, if $Q$ is $k$-neighborly for $1\leq k \leq \floor{\frac{d}{2}}$, then this number is bounded by:
\[h_k(\partial Q) =\binom{n-d-1+k}{k}.\]
\end{lemma}

 \begin{proof}
 Let $T$ be a triangulation of $Q$.
 We want a bound on $f_{d}(T)$.

 We use the following result from McMullen and Walkup \cite[Thm. 2]{MW71}, cited in a modern version in \cite[Thm. 2.6.11]{DRS10}.
 For any $0\leq j \leq d$,
 \[h_j(\partial Q)-h_{j-1}(\partial Q) = h_j(T)-h_{d+1-j}(T),
 \]
 where $\partial Q$ is the boundary simplicial complex of $Q$, of dimension $d-1$, and we take $h_{-1}(\partial Q)=0$.

 Then we have:
 \begin{align*}
     f_{d}(T) &= \sum_{k=0}^{d+1}h_k(T)\\
     &= h_l(\partial Q)-h_{-1}(\partial Q) +\sum_{j=0}^l h_{d+1-j}(T)+\sum_{k=l+1}^{d+1} h_k(T)\\
     &\geq h_l(\partial Q).
 \end{align*}

 The first $h$-coefficients of neighborly polytopes are well known, as they achieve the maximum allowed by the Upper Bound Theorem (\cite[Lemma 2]{McM70}, see also \cite[Lemma 8.26]{Zie95}).
 In particular we have:
 \[
 h_l(\partial Q) = \binom{n-d-1+l}{l}.
 \]
  \end{proof}

\begin{remark}
From the proof one derives that a triangulation $T$ has exactly $h_{k}(\partial(Q))$ cells if, and only if, $h_j(T)=0$ for every $j\ge  k+1$.
This, in turn, is equivalent to all interior cells of $T$  having dimension at least $d-k$.
\end{remark}

As a consequence of the previous two lemmas we have:

\begin{theorem}\label{thm:induction_regtriang}
Let $P=(p_1, \ldots, p_{n-1}, q)$ be a configuration of $n$ points in convex position in $\mathbb{R}^d$ such that:
\begin{enumerate}[(i)]
\item for every $d+1\leq i \leq n-1$, $p_{i}$ and $q$ are triangulation-inseparable in $P_i:=(p_1, \ldots, p_i, q)$, and
\item the point configuration $P/q$ is $k$-neighborly.
\end{enumerate}
Then \[|\regtriang{P}| \geq \prod_{m=d}^{n-1} \binom{m-d+k}{k},\]
which is of order $(n!)^{k\pm o(1)}$.

\end{theorem}

\begin{proof}
We proceed by induction on $n$. In the base case $n=d+1$ we have that
$\pc=\pc_d$ is a simplex, with only one regular triangulation, so the result is trivial.

Suppose that we have the result until $\pc_{m-1}$. The first hypothesis implies that $\p_{m}$ and $\p[q]$ are triangulation-inseparable in $\pc_{m}$.
It follows from Lemma~\ref{lem:inductionstep} that
\[|\regtriang{\pc_{m}}| \geq |\regtriang{\pc_{m-1}}|\times (C_m+1),\]
where $C_m$ is the minimum number of cells in a regular triangulation of $\pc_{m} /q=(\pc /q)\setminus\{\p_{m+1}, \ldots, \p_{n-1}\}$. Since $\pc/q$ is $k$-neighborly and a subset of a $k$-neighborly point configuration is still $k$-neighborly, we know that $\pc_{m} /q$ is a $k$-neighborly $(d-1)$-dimensional simplicial polytope on $m$ vertices.
Hence, Lemma~\ref{lem:cells_triang_neighb} implies that $C_m\geq \binom{m-d+k}{k}$.

At the end:
\begin{align*}
|\regtriang{\pc}|
&\geq \prod_{m=d}^{n-1} \binom{m-d+k}{k}\\
&\geq \prod_{m=d}^{n-1} \left( \frac{m-d+k}{k}\right)^{k}\\
&= \left((n-d-1+k)!\right)^{k}\times \frac{1}{\left((k-1)! k^{(n-d)}\right)^{k}} \\
&= \exp(kn\log n + o(n \log n)).
\qedhere
\end{align*}
\end{proof}

The combination of these results provides Theorem~\ref{thm:manytriangulations}: a lower bound of order $(n!)^{\ffloor{d-1}{2} \pm o(1)}$ for the number of regular triangulations of cyclic polytopes in certain realizations.
Recall that the \defn{cyclic $d$-polytope} with $n$ vertices is a neighborly simplicial polytope that can be realized as the convex hull of $n$ arbitrary points $p_1,\dots,p_n$ along the moment curve $\set{(t,t^2,\dots,t^d)\in \mathbb{R}^d}{t\in \mathbb{R}}$. See for example \cite{Zie95} for details.

\begin{proof}[Proof of Theorem \ref{thm:manytriangulations}]
We first fix the last vertex $p_n$ on the moment curve and then define the points $p_1, \ldots, p_{n-1}$ consecutively.
At step $i$, we slide the point $p_i$ along the moment curve until it is close enough to $p_n$ so that Lemma~\ref{lem:triang_insep} implies them to be triangulation-inseparable, after a perturbation of $p_i$ into very general position if needed. The vertex figure of the last vertex in a cyclic polytope is  a $(d-1)$-dimensional cyclic polytope, and therefore $\ffloor{d-1}{2}$-neighborly. Hence Theorem~\ref{thm:induction_regtriang} gives the result.
\end{proof}

It is not clear to us whether cyclic polytopes (or neighborly polytopes in general) do indeed have more triangulations than ``typical'' simplicial polytopes of the same dimension and number of vertices. In fact, in dimension two quite the opposite is true: the convex $n$-gon minimizes the number of triangulations and of regular triangulations among point configurations of $n$ points in general position~\cite{KupavskiiVolostnovYarovikov21,GonzalezSantos21}.

\section{Many polytopes}\label{sec:manypolytopes}

Let us call \defn{polytopal (simplicial) $d$-ball} any (labeled) simplicial complex that can be realized as a regular triangulation of a configuration of points in dimension $d$. By adding a point ``at infinity'' to a polytopal $d$-ball one obtains a polytopal $d$-sphere with one more vertex, and viceversa. Thus,
the number of combinatorially different labeled polytopal $d$-balls with $n$ vertices coincides with the number of combinatorially different labeled simplicial $(d+1)$-polytopes with $n+1$ vertices.

On the other hand, if two simplicial polytopes are combinatorially different  then no triangulation of the first can be combinatorially equal to one of the second, because we can recover the boundary complex of a simplicial polytope from any of its triangulations. Hence:

\begin{lemma}
\label{lem:triangs_to_polytopes}
If $\pc_1, \dots, \pc_N$ are configurations of dimension $d$ and size $n$  in convex and general position and with combinatorially different convex hulls, then there are at least
\[
\sum_{i=1}^N |\regtriang{\pc_i}|
\]
combinatorially different labeled simplicial $(d+1)$-polytopes with $n+1$ vertices.
\end{lemma}

In this section we show that not only cyclic polytopes but all the \emph{Gale sewn} polytopes introduced in~\cite{Pad13}  fulfill (in certain realizations) the conditions of Theorem~\ref{thm:induction_regtriang}. This provides us with a large family of polytopes with many regular triangulations, to which we can apply Lemma~\ref{lem:triangs_to_polytopes} and obtain even more polytopes.

In order to have a self-contained presentation, we give in  Section~\ref{sub:lextechnical} all the definitions and lemmas that are used in the proofs of the constructions in the Section~\ref{sub:manypolytopes}. Most of the contents of the latter can be traced back to \cite{Pad13,GP16}, but observe that the presentation in~\cite{Pad13} is formulated in the Gale dual setting of extensions, while ours, and the one in \cite{GP16} is already formulated in a primal setting of liftings.

\subsection{Lexicographic liftings}
\label{sub:lextechnical}

A central tool for our  construction are lexicographic liftings, which are a way to derive $(d+1)$-dimensional point configurations from $d$-dimensional point configurations.

\begin{definition}\label{def:lexicographiclifting}
A \defn{positive lexicographic lifting} of a point configuration $P=(p_1,\dots,p_n)\subset\mathbb{R}^d$ (with respect to the order induced by the labels) is any configuration $\widehat P=(\widehat p_1,\dots,\widehat p_n,\widehat q)$ of $n+1$ labeled points in~$\mathbb{R}^{d+1}$ such that:
\begin{enumerate}[(i)]
\item $\widehat q$ is a point in the  halfspace $x_{d+1}>0$,
 \item for $1\leq i\leq n$, the point $\widehat p_i$ lies in the half-line  from $\widehat q$ through $(p_i,0)$,
 \item \label{it:hyperplanes} for $d+2\leq i\leq n$, and for every hyperplane $H$ spanned by $d+1$ points taken among $\left\{\widehat p_{{1}},\dots,\widehat p_{{i-1}}\right\}$,
 the points $\widehat q$ and $\widehat p_i$ lie on the same side of $H$.
\end{enumerate}
\end{definition}

\begin{remark}\label{rmk:epsilon_lex_lift}
Positive lexicographic liftings exist for every point configuration, and are a special case of the lexicographic liftings produced with a sign vector in $\{+, -\}^n$, as defined e.g.\ in \cite[Def. 4.1]{GP16}.
One way to construct a positive lexicographic lifting is to choose $\widehat q$ arbitrarily with $x_{d+1} >0$ and then take $\widehat p_i := (1-\varepsilon_i) \widehat q + \varepsilon_i (p_i,0)$ for constants $0 < \varepsilon_n \ll \varepsilon_{n-1} \ll\dots \ll  \varepsilon_1$. See Figure~\ref{fig:lifting}.
\end{remark}

\begin{figure}[ht]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=.9\linewidth]{Figures/lifting1}
\caption{$P$}
\label{fig:lifting1}
\end{subfigure}\quad
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=.9\linewidth]{Figures/lifting2}
\caption{$\widehat P$}\label{fig:lifting2}
\end{subfigure}\quad
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=.9\linewidth]{Figures/lifting3}
\caption{Checking \ref{it:hyperplanes}}\label{fig:lifting3}
\end{subfigure}
\caption{A positive lexicographic lifting $\widehat P\subset \mathbb{R}^2$ of a configuration $P\subset \mathbb{R}^1$.}
\label{fig:lifting}
\end{figure}

The faces of $\operatorname{conv}(\widehat P)$ that do not contain $\widehat q$ give a particular subdivision of $P$ that is called the \defn{placing}, or \defn{pushing}, triangulation. We refer the reader to Section 4.3.1 of \cite{DRS10} for more details.

\begin{definition}
A face $F$ of a polytope $Q$ is \defn{visible} from a point $p\in \mathbb{R}^d$ if there is an affine functional that is zero on $F$, strictly positive on $p$ and strictly negative on $Q\setminus F$.
$F$ is \defn{hidden} from $p$ if there is an affine functional that is zero on $F$ and strictly negative both on $p$ and on $Q\setminus F$.
Note that a face that is not a facet can be both visible and hidden from $p$, and if $p$ is in general position with respect to $Q$ and $p\notin Q$, then any face of $Q$ (even facets) is either visible or hidden from $p$.

Let $\pc=(\p_1, \ldots, \p_n)$ be a point configuration in general position in $\mathbb{R}^d$. We denote $P_i:=(\p_1, \ldots, \p_i)$.
The \defn{placing triangulation} $\tr_n$ of $\pc_n$ is defined iteratively by taking for $\tr_1$ the singleton $\{1\}$ and for $\tr_i$ the union of the faces of $\tr_{i-1}$ with all simplices of the form $F\cup\{i\}$ where $F$ gives a face of $\operatorname{conv}(\pc_{i-1})$ that is visible from $\p_i$.
$\tr_i$ is the only triangulation of $\pc_i$ that contains $\tr_{i-1}$.
The $\defn{pulling}$ triangulation of $\pc$ is the union of all simplices that give proper faces of $\operatorname{conv}(\pc)$ and all $F\cup\{n\}$ where $F \subseteq [n-1]$ gives a proper face of $\operatorname{conv}(\pc)$.
(Proper faces are those different from the whole polytope).

\end{definition}

\begin{lemma}\label{lem:faces_lexlifting}
Let $\widehat P=(\widehat p_1,\dots,\widehat p_n,\widehat q)$ be a positive lexicographic lifting of the point configuration $P=(p_1,\dots,p_n)\subset\mathbb{R}^d$ in convex position.
For $i\in [n]$ we denote $P_i := (p_1, \ldots, p_i)$ and $\widehat P_i := (\widehat p_1, \ldots, \widehat p_i)$.
Then:
\begin{enumerate}[(i)]
\item\label{it:lexlifting_faces_hidden} The faces of $\operatorname{conv}(\widehat P_n)$ that are hidden from $\widehat q$ are exactly the liftings of faces of the placing triangulation of $P_n$.
\item \label{it:lexlifting_faces_visible} The faces of $\operatorname{conv}(\widehat P_n)$ that are visible from $\widehat q$ are exactly the liftings of faces of the pulling triangulation of $P_n$.
\item\label{it:lexlifting_faces_intermediate} For $i\in [n-1]$, the faces of $\operatorname{conv}(\widehat P_i)$ that are hidden, resp. visible, from $\widehat p_{i+1}$ coincide with the faces that are hidden, resp. visible, from $\widehat q$.
\item \label{it:lexlifting_faces_q} The faces of $\operatorname{conv}(\widehat P)$ are exactly the faces of $\operatorname{conv}(\widehat P_n)$ that are hidden from $\widehat q$, which are the liftings of faces of the placing triangulation of $P_n$, and all $\operatorname{conv}(\set{\widehat p_i}{i\in F}\cup\{\widehat q\})$ where $F$ gives a face of $\operatorname{conv}(P_n)$.
\end{enumerate}
\end{lemma}

\begin{proof}

Items \ref{it:lexlifting_faces_hidden} and \ref{it:lexlifting_faces_visible} are reformulations of \cite[Lemma 4.3.4]{DRS10} and \cite[Lemma 4.3.6]{DRS10}, which correspond to the case where the point $\widehat q$ is ``at infinity''.
In that case, the faces of $\operatorname{conv}(\widehat P_n)$ hidden from $\widehat q$ correspond to the  lower faces of $\operatorname{conv}(\widehat P_n)$, thus to faces of the corresponding induced regular subdivision of $P_n$.
The faces of $\operatorname{conv}(P_n)$ visible from $\widehat q$ correspond to the lower faces of the lifting of $P_n$ induced by the opposite (negative) heights.
Then, in both \cite[Lemma 4.3.4]{DRS10} and \cite[Lemma 4.3.6]{DRS10} where we take the opposite heights, the condition on the constant $c_0$ and the heights amounts to asking that the lifting is a positive lexicographic lifting.

Item \ref{it:lexlifting_faces_intermediate} follows from the definitions and the fact that a face of a polytope $Q$ is hidden, resp. visible, from a point $p$ if and only if it is contained in a facet of $Q$ that is hidden, resp. visible from $p$.

For \ref{it:lexlifting_faces_q}, notice that the faces of $\operatorname{conv}(\widehat P)$ that do not contain $\widehat q$ are exactly the faces of $\operatorname{conv}(\widehat P_n)$ hidden from $\widehat q$. The same argument as before shows that they form the placing triangulation of $\widehat P_n$. If $F\subseteq [n]$ is such that $F\cup\{\widehat q\}$ gives a face of $\widehat P$, let $h$ be a supporting hyperplane of this face. Then the intersection of $h$ with the hyperplane $w_{d+1}=0$ is a supporting hyperplane of the face given by $F$ for $\operatorname{conv}(P_n)$ in~$\mathbb{R}^d\times\{0\}$.
\end{proof}

\begin{corollary}\label{cor:combitypelift}
Let $P=(p_1, \dots, p_n)\subset \mathbb{R}^d$ be a point configuration in convex position.
Let $\widehat P=(\widehat p_1,\dots,\widehat p_n,\widehat p_{n+1})$
 be a positive lexicographic lifting of $P$ and let $\wwh{P}=(\wwh p_1 ,\dots,\wwh p_n,\wwh p_{n+1}, \wwh p_{n+2})$
be a positive lexicographic lifting of $\widehat P$, with respect to the same order. Then
the combinatorial type of $\operatorname{conv}(\wwh P)$ is completely determined by (the oriented matroid of) the point configuration~$P$.
\end{corollary}

\begin{proof}
According to Lemma \ref{lem:faces_lexlifting} \ref{it:lexlifting_faces_q}, the faces of $\operatorname{conv}(\wwh P)$ are the liftings of faces of the placing triangulation of $\widehat P_{n+1}$ and all $\operatorname{conv}(\set{\wwh p_i}{i\in F}\cup\{\wwh p_{n+2}\})$ where $F$ gives a face of $\operatorname{conv}(\widehat P_{n+1})$.
The definition of the placing triangulation and Lemma \ref{lem:faces_lexlifting} \ref{it:lexlifting_faces_hidden}, \ref{it:lexlifting_faces_visible}, \ref{it:lexlifting_faces_intermediate} imply that the placing triangulation of $\widehat P_{n+1}$ (and thus also the faces of $\operatorname{conv}(\widehat P_{n+1})$) is determined by the placing and pulling triangulations of the $P_i$.
\end{proof}

If one starts with a $0$-dimensional point configuration (that is a point repeated multiple times), and then perfoms a sequence of positive lexicographic liftings always with respect to the same order, then one obtains a cyclic polytope. If the order is altered at each step, then many combinatorial types of polytopes are obtained, but not necessarily neighborly.
Moreover, different lifting orders might give rise to equivalent polytopes. However, if one restricts to changing the order of the lifting only every two dimensions, then neighborliness is preserved and the combinatorial type can be controlled.
This is used in~\cite{Pad13} to construct many neighborly polytopes. The original presentation in \cite{Pad13} is in terms of lexicographic extensions of the Gale dual, but we refer to the following primal version for liftings taken from~\cite{GP16}. We repeat the main ideas of that proof for the reader's convenience.

\begin{theorem}[{\cite[Theorem 5.5(i)]{GP16}}]\label{thm:neighborlylift}
 Let $P=(p_1, \dots, p_n)\subset \mathbb{R}^d$ be a $k$-neighborly point configuration in general position. Let $\widehat P=(\widehat p_1,\dots,\widehat p_n,\widehat p_{n+1})$
 be a positive lexicographic lifting of $P$ and let $\wwh{P} =(\wwh p_1,\dots,\wwh p_n,\wwh p_{n+1}, \wwh p_{n+2})$
be a positive lexicographic lifting of $\widehat P$, with respect to the same order. Then
$\wwh P$ is $(k+1)$-neighborly.
\end{theorem}

\begin{proof}
First, we prove that $\widehat P$ is $k$-neighborly. Let $S$ be a subset of $[n]$ of size $k$ or $k-1$. Then $\set{p_i}{i\in S}$ is the vertex set of a face of $\operatorname{conv}(P)$. Hence, it follows from Lemma~\ref{lem:faces_lexlifting} \ref{it:lexlifting_faces_q} that $\set{\widehat p_i}{i\in S}\cup\{\widehat p_{n+1}\}$ is the vertex set of a face of $\operatorname{conv}(\widehat P)$.

The same reasoning shows that any subset of $\wwh P$ of size $k+1$ that contains $\wwh p_{n+2}$ or $\wwh p_{n+1}$ is the vertex set of a face of $\wwh P$. We deal with the remaining cases by induction on $n$.
Let $S$ be a subset of $[n]$ of size $k+1$. We want to show that $\set{\wwh p_i}{i\in S}$ is the vertex set of a face of $\operatorname{conv}(\wwh P)$. If $n=d+1$, then $P$, $\widehat P$ and $\wwh P$ are simplices, so any subset of the points is the vertex set of a face.
If $S$ contains $n$, then we have that $S\setminus\{n\}$ gives a face of $\operatorname{conv}(P)$, thus  $S$ gives a face of the pulling triangulation of $P$, a face of $\widehat P_n$, and a face of the placing triangulation of $\widehat P$. It follows from Lemma \ref{lem:faces_lexlifting} \ref{it:lexlifting_faces_q} that $S$ gives a face of $\wwh P$. Finally, if $S$ does not contain $n$, we can use the induction on $n-1$.
\end{proof}

The following lemma allows us to prove that the combinatorial type can be controlled without explicitly using the rigidity of neighborly oriented matroids of odd rank as it was originally done in \cite[Proposition~6.7]{Pad13}.

\begin{lemma}\label{lem:typecontrolled}
 Let $P=(p_1, \dots, p_{n})$ be an $r$-neighborly point configuration in even dimension $d=2r$ such that $n>d+2$. Let $\widehat P=(\widehat p_1,\dots,\widehat p_{n},\widehat p_{n+1})$ be a lexicographic lifting of $P$ and let $\wwh{P}=(\wwh p_1,\dots,\wwh{p}_{n},\wwh p_{n+1}, \wwh p_{n+2})$ be a positive lexicographic lifting of~$\widehat P$, with respect to the same order.

Then $n$ is the only index $k\in [n]$ such that the double contraction $\wwh{P}/\{\wwh p_{n+1}, \wwh p_k\}$ is $r$-neighborly.

\end{lemma}

\begin{proof}
For $r\geq 1$, we know that $\wwh P$ is $2$-neighborly, so all pairs $\{n+1, k\}$ for $k\in [n]$ give edges of $\wwh P$, and all points of the configuration $\wwh P/\{\wwh p_{n+1}\}$ are vertices. This justifies that the double contraction $\wwh{P}/\{\wwh p_{n+1}, \wwh p_k\}$ is well-defined.
If $d=r=0$, $\wwh P/\{\wwh p_{n+1}\}$ is a $1$-dimensional configuration of points ordered linearly $n, n-1, \ldots, 2, 1, n+2$. Thus, the double contraction is well-defind only for $k=n+2$ and $k=n$ and we already have the result of the lemma.

Note that $P$ is a realization of $\wwh P/\{\wwh p_{n+2}, \wwh p_{n+1}\}$.
It follows from the definition of contraction that a set $S\subseteq [n]\setminus\{k\}$ gives a face of $\operatorname{conv}(\wwh{P}/\{\wwh p_{n+1}, \wwh p_k\})$ if and only if $S\cup\{n+1, k\}$ gives a face of $\operatorname{conv}(\wwh P)$.

We denote $\widehat P_i:= (\widehat p_1, \ldots, \widehat p_i)$ for $i\in [n]$.

We first show that $\wwh{P}/\{\wwh{p}_{n+1}, \wwh{p}_{n}\}$ is $r$-neighborly.
Let $S\subseteq [n+2]\setminus\{n+1, n\}$ be a subset of cardinality~$r$.
If $S$ contains $n+2$, we define $S':=S\cup\{n\}\setminus\{n+2\}$. $S'$ is a subset of $[n]$ of cardinality $r$, hence it defines a face of $\operatorname{conv}(P)$ and $S\cup\{n+1, n\}=S'\cup\{n+1, n+2\}$ indeed defines a face of $\operatorname{conv}(\wwh P)$.
If $S$ does not contain $n+2$, then it is a subset of $[n]$ of cardinality $r$ and hence it gives a face of $\operatorname{conv}(P)$.
Thus, $S\cup\{n\}$ gives a face of the pulling triangulation of $P$, and by Lemma~\ref{lem:faces_lexlifting}\ref{it:lexlifting_faces_visible} a face of $\operatorname{conv}(\widehat P_n)$ that is visible from $\widehat p_{n+1}$.
Therefore, $S\cup\{n, n+1\}$ gives a face of the placing triangulation of $\operatorname{conv}(\widehat P)$, thus a face of $\operatorname{conv}(\wwh P)$.

Now, let $k$  be an element of $[n-1]$. To show that $\wwh P/\{\wwh p_{n+1}, \wwh p_k\}$ is not $r$-neighborly, we will exhibit a subset $S\subseteq [n]\setminus\{n+1, k\}$ of cardinality $r$ such that $S\cup\{n+1, k\}$ does not give a face of $\operatorname{conv}(\wwh P)$.
Since $n > d+2$, we can find a subset $W$ of $[n]$ of cardinality $d+2=2(r+1)$ that contains $k$ but not $n$.
Radon's theorem implies that there is a partition of $W$ into two subsets $W_1$ and $W_2$ such that $\operatorname{conv}(\set{p_i}{i\in W_1})\cap\operatorname{conv}(\set{p_j}{j\in W_2})\neq \emptyset$.
In particular, $W_1$ and $W_2$ do not give faces of $\operatorname{conv}(P)$.
Since $P$ is $r$-neighborly, $W_1$ and $W_2$ necessarily have at least $r+1$ elements, so they are both exactly of cardinality $r+1$.
(This is where the assumption of even dimension is used).
We define $T$ to be the $W_i$ that contains $k$, and $S:=T\setminus\{k\}$.
Since $T$ does not give a face of $\operatorname{conv}(P)$ and does not contain $n$, it does not give a face of $\widehat P_n$ that is visible from $\widehat p_{n+1}$.
Hence, $T\cup\{n+1\}=S\cup\{n+1, k\}$ does not give a face of the placing triangulation of $\widehat P$.
However, all faces of $\wwh P$ not containing $n+2$ must be faces of the placing triangulation of $\widehat P$ by Lemma~\ref{lem:faces_lexlifting}\ref{it:lexlifting_faces_q}.
Thus, $S\cup\{n+1, k\}$ does not give a face of $\wwh P$.
\end{proof}

\begin{corollary}[{\cite[Proposition 6.1]{Pad13} and \cite[Lemma 6.1]{GP16}}]\label{cor:typecontrolled}
Let $P=(p_1, \dots, p_{n})$ be an $r$-neighborly point configuration in even dimension $d=2r$.
Then there are at least $\frac{n!}{(d+2)!}$ distinct labeled combinatorial types of $(d+2)$-polytopes
with $n+2$ vertices obtained by the following construction:

\begin{itemize}
\item Choose a permutation $\sigma$ of $n$.
\item Define the point configuration $P^\sigma=(p_{\sigma(1)}, \ldots, p_{\sigma(n)})$.
\item Let $\widehat P^\sigma$ be a positive lexicographic lifting of $P^\sigma$ and let $\wwh P^\sigma=(\wwh p_{\sigma(1)}, \ldots, \wwh p_{\sigma(n)}, \wwh p_{n+1}, \wwh p_{n+2})$ be a positive lexicographic lifting of $\widehat P^\sigma$.
\item Define $\wwh P =(\wwh p_1, \ldots, \wwh p_n, \wwh p_{n+1}, \wwh p_{n+2})$.
\item Take the convex hull $\operatorname{conv}(\wwh P)$.
\end{itemize}

\end{corollary}

\begin{remark}
In fact, \cite[Lemma 6.1]{GP16} gives a bound improved by a factor $n+1$, but this does not change the asymptotics of the bound on the total number of polytopes.
\end{remark}

\begin{proof}
Let $\wwh P=(\wwh p_1, \ldots, \wwh p_n, \wwh p_{n+1}, \wwh p_{n+2})$ be a point configuration in $\mathbb{R}^{d+2}$ obtained as in the statement, with a permutation $\sigma$ that we do not know.
We show by induction on $n$ that we can recover $\sigma(n), \sigma(n-1), \ldots, \sigma(d+3)$ from $P$ and the face lattice of $\operatorname{conv}(\wwh P)$. This implies that distinct choices for $\sigma(n), \sigma(n-1), \ldots, \sigma(d+3)$ give distinct labeled combinatorial types $\operatorname{conv}(\wwh P)$,
and there are $\frac{n!}{(d+2)!}$ such choices.
Suppose that we have already recovered $\sigma(n), \sigma(n-1), \ldots, \sigma(m)$ for some $d+3 < m \leq n+1$. We consider the point configuration $\wwh P_{m-1} := \wwh P \setminus\{\wwh p_{\sigma(n)}, \ldots, \wwh p_{\sigma(m)}\}$ (where we abuse notation for the labels but the only important thing is to record the last two points).
It follows from Corollary \ref{cor:combitypelift} that its combinatorial type is well defined, because it is obtained as the relabeling of the double lifting of the point configuration $(p_{\sigma(1)}, \ldots, p_{\sigma(m-1)})$ (with the two additional points $\wwh p_{n+1}$ and $\wwh p_{n+2}$).
Moreover, since the point configuration $(p_{\sigma(1)}, \ldots, p_{\sigma(m-1)})$ is $r$-neighborly, it follows from Lemma~\ref{lem:typecontrolled} that we can recover $\sigma(m-1)$ as the only index $k\in [m-1]$ such that $\wwh P_{m-1}/\{\wwh p_{n+1}, \wwh p_k\}$ is $r$-neighborly.
\end{proof}

\subsection{Construction of many polytopes}
\label{sub:manypolytopes}

We will use this slight variation of the construction used in~\cite{Pad13} to give a lower bound for the number of polytopes.
\begin{theorem}[{\cite[Theorem 6.8]{Pad13}}]\label{thm:manyneighborly}
 The number of labeled combinatorial types of neighborly $d$-polytopes with $n>d$ vertices obtained from a $0$-dimensional point configuration by a sequence of positive lexicographic liftings (with orders that might change along each step of the sequence) is at least
 \[(n!)^{\ffloor{d}{2}\pm o(1)}.\]
\end{theorem}

\begin{proof}
We build iteratively sets $\mathcal{P}_{2k}$ that contain realizations of distinct labeled combinatorial types of neighborly polytopes of dimension $2k$ with $n-d+2k$ vertices.

We define $\mathcal{P}_0$ to be the singleton with the degenerate configuration of $n-d$ labeled points in the $0$- dimensional space.

Suppose that we have constructed $\mathcal{P}_{2k}$ for some $0\leq k < \ffloor{d}{2}$.
Let $\mathcal{P}_{2k+2}$ be the union over all configurations $P\in \mathcal{P}_{2k}$ of the distinct labeled point configurations obtained from $P$ by relabelings and two positive lexicographic liftings in the same order, as in Corollary~\ref{cor:typecontrolled}. This union is disjoint because if $\wwh P$ is a double lifting of $P$, we can recover the combinatorial type of $P$ by taking $\wwh P/\{\wwh p_{n-d+2k+2}, \wwh p_{n-d+2k+1}\}$. Hence, Corollary~\ref{cor:typecontrolled} gives that $|\mathcal{P}_{2k+2}|\geq |\mathcal{P}_{2k}|\times \frac{(n-d+2k)!}{(2k+2)!}$.
Theorem~\ref{thm:neighborlylift} ensures that the point configurations in $\mathcal{P}_{2k+2}$ are neighborly.

For $k=\ffloor{d}{2}$ we obtain that:
\begin{align*}
|\mathcal{P}_{2\ffloor{d}{2}}| & \geq
\prod_{k=0}^{\ffloor{d}{2}-1} \frac{(n-d+2k)!}{(2k+2)!}\\
&\geq \frac{\left( (n-d)! \right)^{\ffloor{d}{2}}}{\prod_{k=1}^{\ffloor{d}{2}} (2k)!}\\
&=(n!)^{\ffloor{d}{2} + o(1)}.
\end{align*}

If $d$ is odd, instead of taking a pyramid as in \cite[Corollary~6.10]{Pad13}, we do one last positive lexicographic lifting on all the elements of $\mathcal{P}_{2\ffloor{d}{2}}$ to obtain $(n!)^{\ffloor{d}{2} + o(1)}$ realizations of distinct labeled combinatorial types of $d$-polytopes with $n$ vertices.
This variant still conserves the number of distinct combinatorial types since we recover the polytopes in $\mathcal{P}_{2\ffloor{d}{2}}$ by taking the vertex figures of the last labeled point.
\end{proof}

The combination of these constructions allows us to prove Theorem~\ref{thm:manypolytopes}: The number of different labeled combinatorial types of $d$-polytopes with $n$ vertices for fixed $d>3$ and $n$ growing to infinity is at least $(n!)^{d-2 \pm o(1)}$.

\begin{proof}[Proof of Theorem~\ref{thm:manypolytopes}]
The last step of the construction of the many $d$-polytopes of Theorem~\ref{thm:manyneighborly} is a positive lexicographic lifting $\widehat \pc = (\widehat p_1, \ldots, \widehat p_{n-1}, \widehat q)$ from a $\floor{\frac{d-1}{2}}$-neighborly $(d-1)$-polytope $P$.

Lemma~\ref{lem:triang_insep} ensures that we can do this lifting step by step so that for every $i$ from $d+1$ to $n-1$, $\widehat p_{i}$ and $\widehat q$ are triangulation-inseparable in $(\widehat p_1, \ldots, \widehat p_i, \widehat q)$. Indeed, the value of $\varepsilon_i$ in Remark~\ref{rmk:epsilon_lex_lift} can be taken arbitrarily small. While very general position is not guaranteed by the construction, note that these configurations are in general position, and hence we can do a small perturbation into very general position if needed without changing the combinatorial type.

Moreover, note that by construction $P$ is the contraction $\widehat \pc /\widehat q$, and that similarly $( p_1, \ldots, p_i) = (\widehat p_1, \ldots, \widehat p_i, \widehat q)/\widehat q$. These contractions are thus $\floor{\frac{d-1}{2}}$-neighborly.

Hence Theorem~\ref{thm:induction_regtriang} applies: each of these polytopes has at least $(n!)^{\ffloor{d-1}{2}n\pm o(1)}$ regular triangulations. Then Lemma~\ref{lem:triangs_to_polytopes} gives us a lower bound of $(n!)^{d-1\pm o(1)}$ labeled simplicial types of $(d+1)$-polytopes with $n$ vertices.
\end{proof}

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
Q: Slow performant query in Arangodb Hypothetically I have two collections.
users (document type)and
relations (edge type)
Relations collection has two attributes {user1, user2} and is indexed combined.
My following query is extremely slow:

FOR r in relations filter (r.user1 == 'xyz' && r.user2 == 'abc') || (r.user1 == 'abc' && r.user2 == 'xyz')
RETURN r

With 800 users and 5000 relationship it takes 35 seconds.

A: I didn't create the indexes properly before. I recreated the indexes again and then the performance was improved.

redpajama_set_name: RedPajamaStackExchange


text:
Ridgeville Corners is an unincorporated community and census-designated place (CDP) in southern Ridgeville Township, Henry County, Ohio, United States. It has a post office with the ZIP code 43555. The population was 435 at the 2010 census.

History
Ridgeville Corners was laid out at an unknown date as an early trading point of Ridgeville Township. The First Congregational Church in Ridgeville Corners was a stop on the Underground Railroad in the mid-1800s. A post office has been in operation at Ridgeville Corners since 1841.

Geography
Ridgeville Corners is located along U.S. Route 6, approximately  northwest of Napoleon, the Henry county seat. It is located  southeast of Bryan.

According to the United States Census Bureau, the CDP has an area of , all of it recorded as land. As its name suggests, Ridgeville Corners sits on high ground, with the northwest side draining towards the Tiffin River, a south-flowing tributary of the Maumee River, and the southeast side of the community draining directly to the Maumee.

Demographics

Gallery

References

Census-designated places in Henry County, Ohio
Census-designated places in Ohio

redpajama_set_name: RedPajamaWikipedia


text:
Major Adriano Visconti di Lampugnano  (11 November 1915 – 29 April 1945) was one of Italy's top flying aces of the Second World War, shooting down between 10 and 26 enemy aircraft. He was awarded with four Medaglia d'argento al Valor Militare and two Medaglia di Bronzo al Valor Militare "in action".

Early years
The son of Count Galeazzo Visconti and Cecilia Dall'Aglio, he was born in Tripoli, Italian Libya. His parents were two Italian colonists who settled in Tripolitania in 1911. Since he was a teenager he displayed a passion for aircraft.

He joined the Italian Royal Air Force (Regia Aeronautica) in October 1936 and completed his pilot training at the Accademia Aeronautica in Caserta.

Career

Visconti graduated on the Breda BA25, and later flew Ro.41s. In 1939 he was commissioned as a Sub-Lieutenant Pilot (Sottotenente Pilota) and posted to the Breda 65 equipped 159ª Squadriglia, 12º Gruppo 50º Stormo, a ground attack unit based at Tobruk.

He was soon briefly posted to 2º Gruppo Aviazione Presidio Coloniale's 23ª Squadriglia for disciplinary reasons, but was allowed to return to his original unit for "heroism under fire": Visconti's Breda Ba 65 had been attacked by three 33 Squadron Gladiators but, because of his skill as a pilot, Visconti and his crew escaped the encounter.  For this action, in addition to being allowed to return to 50º Stormo, Visconti was awarded the first of his Medaglia di Bronzo. From June 1940 until the end of that year he flew continually without leave, and was awarded another Bronze medal  and two Medaglia d'argento.

By January 1941 50º Stormo had taken such heavy losses that the unit was disbanded. Visconti was detached to the 54º Stormo, 7º Gruppo, 76ª Squadriglia, where he learned to fly the Macchi C.200. He then flew the superior Macchi C.202 from the end of 1941 until the Italian Armistice of September 8, 1943.

On 22 December 1941, in a dogfight over Malta, he was credited with a "probable" Hawker Hurricane. His first official air victory was on the 15 June 1942, when he shot down a Bristol Blenheim near the island of Pantelleria. On 13 August 1942 while flying with a wingman in a formation of two C.202s on a reconnaissance flight over Malta looking for a shipping convoy, the Italian aircraft were bounced by four Supermarine Spitfire escorting the ships. Visconti, only flying the Macchi for the second time, was able to shoot down two Spitfires and damage the other two, allowing the other Macchi to complete the reconnaissance mission. For this double air victory, Visconti was awarded a Medaglia d'Argento al Valor Militare (Silver Medal of Military valour).

After the armistice, Visconti continued to fly with the newly formed National Republican Air Force (Aeronautica Nazionale Repubblicana, or ANR) of the Italian Social Republic (Repubblica Sociale Italiana, or RSI). Shortly after joining the ANR, Visconti was promoted to Captain, commanding the 1st Squadron in the 1st Gruppo Caccia. In May 1944, he was promoted to Major.

Until the disbanding of the ANR in 1945, Visconti flew the Macchi C.205 and the Messerschmitt Bf 109G-10. He was credited with his first "kill" with the ANR on 3 January 1944, when, flying a C.205, he downed a P-38 Lightning South of the Piedmontese city of Cuneo.
Visconti was wounded on several occasions, but never seriously.

Although he never claimed more than ten kills some later source credited him with 26. Most of his victories were while flying the Macchi 205V; the last, a P-47 over Lake Garda, on 14 March 1945, was in a Bf 109 G10 designated "3-4" but in reality was only a "claim", since the pilot he met in the head-on dogfight wasn't actually shot down.

In fact, on the 14th of March 1945 Adriano Visconti was shot down, by the USAAF pilot, 2nd/Lt Charles Clark Eddy Jr. in a P-47 he named "Chickenbones" of 346th FS, 350th FG.  They were in a dogfight in the Lake Garda area when Visconti's plane was hit and disabled by Eddy in a frontal attack, Visconti parachuted out at a very low altitude and survived his moderate wounds, while Eddy's P-47 came back safely to his Pisa airbase.  This story can be verified by reading the article "L'UOMO CHE ABBATTE' VISCONTI" (THE MAN WHO SHOT DOWN VISCONTI), in the 3rd of March 1989 issue of the Italian magazine "Aeronautica", which was based on the notes taken at the time by Eddy in his notebook, allowing to retrace the combat and to understand that he was the "adversary" of Visconti (a fact completely unknown to Eddy himself till then).

Death

On April 29, 1945, Visconti surrendered to communist partisans near Malpensa airfield, Milan, only after he was assured that none of the air and ground personnel of his unit would be killed in retaliatory attacks. His safety was assured by the mayor of Milan as well, when he was first taken for interrogation.

Visconti and the other officers were taken on two buses to the barracks of "Savoy Cavalry" (Savoia Cavalleria) in via Vincenzo Monti.  The barracks was then the command of National Republican Guard Intendancy (Intendenza della Guardia Nazionale Repubblicana). There, at 2 p.m., Visconti and Lieutenant (tenente) Valerio Stefanini, his close collaborator, were seemingly taken for a routine  interrogation. They were both shot in the back by a Russian bodyguard of the communist partisan leader "Iso".

The Russian was charged with murder but was later discharged because the crime had occurred before 8 May 1945, the official end of the war in Italy, and was therefore considered an act of war. Visconti and Stefanini were initially hastily buried in the courtyard of the barracks of Savoy Cavalry. In May 1945, a group of fellow aviators and friends, including Giuseppe Robetto, Ugo Diappi, Luigi Botto and Irma Rachelli, arranged for the bodies to be moved to Cimitero Monumentale di Milano.

See also
 List of World War II aces from Italy

Notes

Bibliography
 D'Amico, Ferdinando and Valentini, Gabriele. "L'uomo che abbattè Visconti" - issue n.3 - March 1989 of "JP4 Aeronautica".
 Beale, Nick, D'Amico, Ferdinando and Valentini, Gabriele. Air War Italy, Axis Air Forces from Liberation of Rome to the Surrender. Shrewsbury, UK: Airlife, 1996. .
Massimello, Giovanni and Apostolo, Giorgio. Italian Aces of World War 2; Osprey Aircraft of the Aces 34. Botley, Oxford UK: Osprey Publishing, 2000.
 Pesce, Giuseppe con Giovanni Massimello. Adriano Visconti Asso di guerra. Parma: Albertelli Edizioni speciali s.r.l. 1997

External links

Blohm & Voss BV P.207.03 Single Seat Pusher Fighter, with information about Visconti

1915 births
1945 deaths
Italian Air Force personnel
Italian aviators
Italian World War II flying aces
Italian military personnel killed in World War II
Recipients of the Silver Medal of Military Valor
Adriano
Deaths by firearm in Italy
Libyan people of Italian descent
Libyan emigrants to Italy
People from Tripoli, Libya


text:
Alexis Tsipras: From "anti-imperialism" to US-NATO's most loyal ally
By Nikos Mottas.
Those who follow the political developments in Greece during the last decade have certainly understood the rapid transformation of SYRIZA- the ruling party in today's coalition government- from a small party of the opportunist left into the country's major social democratic power. In this site we have written extensively, in numerous occasions, about the role of SYRIZA a major pillar (alongside the center-right New Democracy party) of the bourgeois political establishment.
In June 2017, we were writing among other things: "During the last seven years, SYRIZA has been proved an extraordinary case of political deception and ideological manipulation. Foreign policy is another field where SYRIZA and its leadership has turned black into white. The supposed 'radical anti-imperialism' of SYRIZA as an opposition party has been fully transformed, thus proving the real, illusionary character of social democracy".
In the last NATO Summit in Brussels, on July 2018, Greece's PM Alexis Tsipras co-signed all the decisions made by the imperialist alliance, thus certifying Athens' willingness to participate further and deeper in the dangerous euroatlantic plans. After all, despite its small size in terms of population, Greece comes 2nd in defense expenditures among NATO member-states!
The official statistics published by NATO are revealing: In 2017 Greece spent 2.36% of GDP (appr. 4.2 billion euros) in "defense expenditures", outrunning countries like the United Kingdom (2.12%), France (1.79%) and Germany (1.24%)! Alexis Tsipras' coalition government spends billions of euros for the military advancement of NATO (that's the truth, no matter if they call it "defense expenditures"), while the Greek working class and the country's popular strata face continuous austerity, antiworker measures, cuts in wages and pensions, high unemployment and heavy taxation.
SYRIZA follows the pattern of the previous governments, those of New Democracy and PASOK. It is characteristic that during the last 7 years of the capitalist economic crisis, all Greek governments maintained military expenditures over 2%, despite any fluctuations. Based on NATO's official statistics, during the last seven years of the crisis, the Greek people paid almost 44 billion dollars (37.5 billion euros) in military expenditures.
The SYRIZA-ANEL coalition government boasts about its "multi-dimensional foreign policy" and about supposedly making Greece a "pillar of stability" in the region. In fact, what the government of Mr. Tsipras does is to deepen the country's involvement in the imperialist competitions, for the sake of Greece's capitalists. Under the leadership of Tsipras, Greece has become the standard-bearer of NATO and EU's interests in the region of east Mediterranean.
Left: 1999, a young Tsipras demonstrating against Clinton's
visit in Athens; Right: 2017, PM Tsipras visits U.S. President
Trump at the White House.
Tsipras himself is a characteristic case of political opportunism and deception. He began his political careers as a radical leftist, with "anti-NATO" views.
As a member of the youth wing of Synaspismos (predecessor of today's SYRIZA) in 1999 he was protesting against Bill Clinton's visit in Athens, while in 2001 he was participating at the large demonstrations against the G8 Summit in Genova. On May 2012, the political programme of SYRIZA was proclaiming among other things: "Our policy includes... the closure of all the foreign military bases in Greece. Immediate abolition of the bases in Souda and Aktio. We want neither Greece in NATO, nor NATO in Greece and we are fighting for its abolition".
In 2014, during a press conference in Thessaloniki, Mr. Tsipras was stating that "in the new international situation there is no reason for NATO's existence" and he was assuring the people that this is a "principle position" that "will not change". Of course, the course of developments showed that Tsipras- like every opportunist- was blatantly lying and that his supposed "anti-imperialism" was a great political deception.
Since he became a Prime Minister, on January 2015, Tsipras has done almost anything in order to prove his loyalty to NATO and the EU. The existing US-NATO military bases in Greece were used for various imperialist attacks in Syria, Iraq and Libya, while new ones (e.g. Alexandroupolis' port) began operating in order to facilitate the imperialist plans. At the same time, Greece never stopped to participate actively in NATO's joint military drills, as well as in bilateral exercises with Israel.
Greece: A "geopolitical hinge" between NATO and the EU in the region
Greece: A "playground" for imperialists.
In many occasions, the U.S. ambassador in Athens Geoffrey Pyatt has praised the role of the Greek government as a stable ally of the US-NATO interests in the region. In one of his interviews, the ambassador called Greece as a "geopolitical hinge" between NATO and the EU, quite useful for the promotion of Washington's plans in the Balkans and east Mediterranean.
The choice of the U.S. as the "honored country" in this year's Thessaloniki International Fair (which begins today, Saturday 8 September) isn't a random one. Monopoly and business groups from the United States have already arrived in Greece's second largest city, in order to present and promote their products.
Yesterday, Alexis Tsipras met with the U.S. Secretary of Commerce Wilbur Ross who heads a high-level delegation from Washington on the occasion of the International Fair. During the meeting, Tsipras underlined the "great importance of the historic relations between our countries and mentioned that the US-Greek relations have also a "very significant perspective, especially in economic relations".
Tsipras' remarks actually verify that his government aims to expand and deepen its collaboration with the imperialist superpower. According to U.S. ambassador Pyatt, the ongoing presence of american monopolies and business groups in Thessaloniki consists "the most important activity of the U.S. in northern Greece since the Marshall Plan" and covers a wide range of the Greek-American relations. Ambassador Pyatt said that Washington regards Greece as a "regional pillar" for the U.S. intervention in Western Balkans and praised the role of the Greek government in the "Prespes Agreement" with FYROM.
The U.S. government could not find a better collaborator for the promotion of its interests than Mr. Tsipras and his government. Especially in a period when the traditional alliance between U.S. with Turkey is facing a serious crisis, the SYRIZA-ANEL government proves its servile- towards the imperialists- political orientation.
As for Alexis Tsipras himself, he will remain in the history of politics as a deplorable example of a "left-winger" who did the "dirty job" for the bourgeois class and served the imperialists better than any right-wing politician could ever do.
* Nikos Mottas is the Editor-in-Chief of 'In Defense of Communism'.
Tags EU, Geoffrey Pyatt, Greece, Imperialism, NATO, Nikos Mottas, SYRIZA, SYRIZA and the bourgeoisie, Trump, Tsipras, USA
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A ginger cat who had the most sorrowful eyes on his face and a difficult life filled with tribulation and pain, was turned around by love. After a long time waiting, he finally found his smile!
A five-year-old ginger cat found his way into the arms of his rescuers at Lollypop Farm (Humane Society of Greater Rochester) but he was emaciated, scared for his life and needed a lot of help and TLC.
"For a while, it seemed as though Nutmeg might have been the saddest ginger cat in the world. He came to Lollypop Farm in September of 2016 as a thin stray with a serious nasal infection and a perpetually sad face. ," the rescue group said.
The ginger feline fought against all the unpredictability living in the wild and survived, but he was left with permanent sadness on his face.
"His ears were tattered, and his face bore the evidence of a difficult life," Lollipop Farm said.
The road to recovery was long, but everyone at Lollipop Farm was in it to help the little sweet guy get back on his paws. He spent some time in an administrative office getting care from the staff who also helped him learn to trust and love again.
When Nutmeg was well enough, he hung out with other rescue kitties in the ginger cat adoption center to learn to socialize before heading to a foster home.
It was then Nutmeg began the wait for his forever human who would love that sad little face with all their hearts and give him a chance for a better life.
"My girlfriend dragged me into Lollipop Farm after seeing him online! I definitely fell in love with his face at first sight, how can you not," Tyler Stadt told.
After a few back scratches, Nutmeg started purring up a storm as if he was telling Tyler that he was ready to go home with them.
Nutmeg's sad eyes will probably stay forever, but the little guy is sad no more. He has found his forever home, and now, he's the happiest he can be!
"Once the saddest cat in the shelter, Nutmeg now had the teeniest, sweetest smile," Lollipop Farm said.


text:
The Hawks Beat A No Magic Orlando
Post category:Game Review / Magic News Roundup
Like pulling off a band-aid I'm getting this over with quickly: the Hawks beat a no magic Orlando by 130-109, a score that flattered Orlando. For a Magic fan it was a hard watch as the defence faltered and Trey Young got his way all evening.
The Orlando Magic had no answer to the hot shooting of the Atlanta Hawks. At the half Atlanta had a 19 point lead, which they easily extended further. Orlando are better than that performance. The Hawks shot 50% from the field over all and 54.5% from the three. That was the killer statistic as turnovers, rebounding, blocks and steals were all relatively well matched.
Young with 22 points did not lead the Hawks in scoring, that honour went to Bogdan Bogdanovic with 23 points. They were amongst the 7 Atlanta players who had double-digit scoring.
Cole Anthony was pretty efficient offensively. He led the team with 23 points and 5 assists.
Jalen Suggs played again at pace all night. He push the ball fast down court whenever it was in his hand. He contributed 14 points, 4 rebounds, 4 assists and a steal.
Wendell Carter Jr finished with 10 points and 10 rebounds for his 25th double-doubles on the season so far. He also added 6 assists.
Franz Wagner got the full focus of the Atlanta defence. His 12 points was his 50th double figure scoring game 60 played.
Mo Bamba recorded 11 points.
Magic Starting Lineup – Cole Anthony (PG), Jalen Suggs (SG), Franz Wagner (SF), Wendell Carter Jr (PF), Mo Bamba (C)
Magic Bench Unit – Gary Harris, Chuma Okeke, Terrence Ross, Admiral Schofield, Ignas Brazdeikis, Robin Lopez (DNP)
The Magic Season
Orlando has a record of 13 wins and 47 losses, dropping back to last in the Eastern Conference. Over the last 10 games played the Magic have won 3, losing the last 4 straight. The home record stands at 5 wins of 26 games played, whilst on the road it is 8 wins in 34 played. In points per game the Magic are scoring 103.3 whilst conceding 112. The efficiency rating for Orlando remains 28th on offence and dropped to 23rd defensively.
Moving On After The Hawks Beat A No Magic Orlando
We are into the All-Star Weekend break now for Orlando.
Cole Anthony, Jalen Suggs and Franz Wagner all travel to Cleveland to take part in the Rising Star games. Cole is also part of the Slam Dunk contest. There will be a post about All Star and Magic involvement coming out soon.
Once the NBA season resumes Orlando will play the Houston Rockets on Friday 25th February in the Amway Center. Houston are last in the Western Conference with a 15-42.
If you want to read Josh Cohen's game review after the Hawks beat a no magic Orlando it is available on the Orlando Magic website.
Tags: Atlanta Hawks, Cole Anthony, Franz Wagner, Houston Rockets, Jalen Suggs, Mo Bamba, Orlando Magic, Wendell Carter Jr
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Orlando Leave New Orleans With The W
GAME DAY 17 – CHARLOTTE VISIT THE AMWAY
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