Convex polytopes are the convex hulls of a finite set of points in Euclidean spaces. They have rich combinatorial, arithmetic, and metrical theory, and are related to toric varieties and to linear programming

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### Polygon of convex arcs

Convex polygons in the plane $R^2$ arise in linear programming where the constraints are linear. The objective linear function attains its maximum at a vertex of the feasible region(if exists).
Assume ...

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### Minkowski sum of polytopes from their facet normals and volumes

By Minkowski's work in the early 1900s, every polytope $P\subset\mathbb R^n$ is determined up to translation by its unit facet normals $u_1,\dots,u_k$ and facet volumes $\alpha_1,\dots,\alpha_k$. ...

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### Possible volumes of lattice polytopes

All polytopes here are assumed to be convex lattice polytopes.
Given a polytope $P$, set $$v(P):= (\operatorname{vol}(F))_{F\text{ a face of }P},$$ where the volume of a $d$-dimensional polytope $P\...

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### The set of polytopes with given $f$-vector

Let $f=(f_0,\ldots f_n)$ be a vector in $\Bbb N^{n+1}$. Let $X$ be the set of all (ordered) $f_0$-tuples in $\Bbb R^n$ whose convex hull has $f$ as its $f$-vector. Assume that $X$ is non-empty. Is ...

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### Projecting a polyhedral cone onto its intersection with the infinity-norm ball

For a point in a convex polyhedral cone, $x\in \mathcal{C} = \{\sum_{i=1}^m \alpha_i r_i \vert \alpha_i \geq 0, r_i \in \mathbb{R}^n \}$, is there an efficient algorithm to project $x$ onto the ...

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### Tiling with Horn's polytopes

Let $n\ge2$ be an integer. Consider the hyperplane $H_n$ of ${\mathbb R}^n$ defined by the equation $x_1+\cdots+x_n=0$ and then the sector $P_n\subset H_n$ defined by the inequalities $x_1\le\cdots\le ...

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### What is the probability distribution of the $k$th largest coordinate chosen over a simplex?

Suppose we're selecting points uniformly at random from the $N$-simplex
$S_N = \{x \in \mathbb R^{N+1}: $ all $ x_i \ge 0$ and $x_1 + \ldots x_N = 1\}$.
One way to do this in practice is choose $N-...

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### condition on rational polyhedral cone to guarantee dual cone is homogeneous

Let $\sigma\subseteq \Bbb R^d$ be a full-dimensional rational polyhedral cone which is strongly convex (i.e. $\sigma\cap-\sigma=0$).
Definition. The cone $\sigma$ is homogeneous if there are ...

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### Can the graph of a symmetric polytope have more symmetries than the polytope itself?

I consider convex polytopes $P\subseteq\Bbb R^d$ (convex hull of finitely many points) which are arc-transitive, i.e. where the automorphism group acts transitively on the 1-flags (incident vertex-...

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### Can computers take uniform samples from a polytope?

For each $r \in \mathbb N $ write $\mathbb Z/ 10^r = \{a/10^r: a \in \mathbb Z\}$ and $P(r)$ for the lattice $(\mathbb Z/10^r)^N \subset \mathbb R^N$.
Suppose the plane $P \subset \mathbb R^N$ is ...

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### Maximal edge length of symmetric polytopes

For me, a polytope is the convex hull of finitely many points. It is said to be vertex-transitive / edge-transitive if its symmetry group acts transitively on its vertices / edges. Let's call a ...

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### Separation of two pointed polyhedral cones using hyperplanes generated by facets

Let $C_1$ and $C_2$ two pointed (that is, with vertex in $0$) polyhedral cones in $\mathbb{R}^n$ with $\dim(C_1)=\dim(C_2)=n$. If
$$\mbox{relative interior}(C_1)\cap \mbox{relative interior}(C_2)=\...

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### Mathematical Structure and Objects Induced by Pairs of Disjoint Subsets

Let $\mathcal{S}$ be a finite, discrete and non-empty set, i.e.,
$$\begin{align} \operatorname{card}\left(\mathcal{S}\right) & =:n\in\mathbb{N}^+\\ V& :=\{v\subset\mathcal{S}\ |\ v\ne\...

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### Fast projection onto a subspace

Given an $n$-dimensional vector $\mathbf{c}\in [0,1]^n$, let $\Delta_{\mathbf{c}}$ be the set of points $\{\mathbf{x}\in [0,1]^n: \langle \mathbf{c},\mathbf{x} \rangle \le 1\}$, where $\langle \mathbf{...

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### Can projecting a simplex onto orthogonal subspaces exposes the same vertices and edges?

Given the regular $n$-dimensional simplex $S\subset\Bbb R^n$ with $n\ge 4$, as well as two orthogonal subspaces $V,W\subset\Bbb R^n$ of dimension $\ge2$ (not necessarily of same dimension, not ...