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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Can the GUE be thought of as a uniform point in a high-dimensional polytope

I have thought about this question for a long time and could only find partial answers. The Gaussian Unitary Ensemble (or GUE) is the eigenvalues of a random Hermitian matrix with complex Gaussian ...
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122 views

3D models of the unfoldings of the hypercube?

There are (apparently) 261 distinct unfoldings of the 4D hypercube, a.k.a., the tesseract, into 3D.1 These unfoldings (or "nets") are analogous to the 11 unfoldings of the 3D cube into the plane.2 ...
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41 views

max volume of inscribed simplex in a ball

Let $B_n$ be the $n$-dimensional unit ball, and $B_n(\epsilon)$ be the spherical cap with height $\epsilon$ I am interested in the quantity $$\Gamma:=\sup_{\Delta:\textrm{ inscribed simplex in ...
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38 views

Conditions on probability measure that generates non-void random polytope

Let $C$ be a non-void compact convex set in $\mathbb{R}^d$, and $\nu$ a probability measure on $C$. Then under what conditions on $C$ and $\nu$, the following statement is true: If ...
6
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1answer
69 views

two-dimensional sections of polyhedral cones

Given a polyhedral cone, its intersection with any two-dimensional plane is either a polygon or a region enclosed by a polygonal curve. Is it a characterization of polyhedral cones? Does there exists ...
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Is every planar point set be projections of vertices of a neighborly 4-polytope?

More exactly, written in coordinates, I'm curious to know if for every point set $(x_i,y_i)$ there are $(x_i,y_i,z_i,w_i)$ be vertices of a neighborly polytope. This problem comes from a simple ...
3
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108 views

Looking for the correct version of a wrong statement from Barvinok's book on convex polyhedra

The book I'm concerned with is "Integer Points in Polyhedra" by A. Barvinok, which, I must say, is turning out to be highly fascinating. A real finite-dimensional vector space $V$ defines the ...
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1answer
60 views

What's the most efficient way to solve this euclidean projection on non-negative affine space constraint?

I've come across this convex optimization problem in my research where I need to project a matrix $X_0$ onto a non-negative affine space constraint and box constraints. Concretely, $X \in ...
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1answer
82 views

Is mean width a Dehn invariant?

Let $P \subset \mathbf{R}^3$ be a convex polyhedron. Let $rP$ be the dilation of $P$ by the positive real number $r$. The Dehn invariant of $P$ is an element of the weird real vector space ...
2
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1answer
40 views

Sampling point uniformly at random satisfying equality constraints

First of all, I apologize in advance if the question has already been asked in some way on this site and/or if there is a widely known solution to this problem. The description of my problem is ...
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20 views

Looking for a homogeneous function with some properties

I'm looking for a 1-homogeneous function $\pi \colon \mathbb{R}^n_{\geq 0} \to \mathbb{R}$ satisfying the following properties: 1) $\pi$ is not concave. This is equivalent to the fact that there ...
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40 views

Number of faces of polytope projecting to lower dimensional polyhedron

Denote $K=\mathrm{conv}(v_1, \ldots, v_n)\subsetneq\Bbb R^m$ to be convex set spanned by vectors $v_i\in\Bbb R^m$ with $m\leq n$ then what technique could be useful to upper bound minimum number of ...
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1answer
302 views

Examples of Polyhedra with Large Shadows

Let $P \subseteq \mathbb{R}^n$ be a polyhedron described by $\mathcal{O}(n^{c_1})$ inequalities, where $c_1$ is a constant. Moreover, let $M\colon P \to \mathbb{R}^2$ be a linear mapping. I'm looking ...
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4answers
462 views

What is the number of equitriangulations of the n-cube?

I wonder if this question has been considered before and if anything is known. My search attempts have failed so far. Let's consider the n-dimesnional cube, $[0,1]^n$, and let's call a simplex with ...
5
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1answer
147 views

Non-closed geodesics on a convex polyhedron in $\mathbb{R}^3$

Let $P$ be the surface of a closed convex polyhedron in $\mathbb{R}^3$. Q. Does every non-closed geodesic $\gamma$ fill $P$ densely? Of course $\gamma$ cannot pass through a vertex of $P$, but ...
7
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1answer
169 views

A polytope with a bound on the sum of any $k$ variables

Let $2\le k\le n-1$ and define the polytope $$P_k(n) = \lbrace (x_1,\ldots,x_n) \in \mathbb{R}^n : -1\le x_{i_1}+\cdots +x_{i_k} \le 1 \text{ for all } 1\le i_1\lt\cdots\lt i_k\le n\rbrace.$$ There ...
4
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2answers
230 views

Marked chain polytope, has this been studied?

Fix $n$ and consider the polytope given by the inequalities $$x_i\leq x_j, \text{ and } 0 \leq x_i \leq a_i \text{ for all } 1\leq i<j \leq n,$$ where $a_i \leq a_i\leq \dots \leq a_n$ are fixed ...
2
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0answers
163 views

Dehn-Sommerville relations for $\Delta$-complexes

Let $M$ be a closed, triangulated manifold of dimension $m$ and $K(M)$ be its triangulation. Let $f_i$ denote the number of $i$-simplices of $K(M)$. As proved by Klee the face numbers satisfy the ...
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27 views

continuous extensions of concave functions

Let $N$ be a lattice. For a ring R we denote $N_R := N \otimes R$. My question is the following: Does a continuous and concave function \begin{eqnarray*} f: N_{\mathbb{Q}} \to \mathbb{R} ...
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1answer
89 views

Linear map of Zonotopes [closed]

Consider a linear system with a map $A: \mathbb{R}^n \rightarrow \mathbb{R}^m$ such that $y = A x$ with $n \geq m$ The input space $x$ is constrained by a zonotope set $\mathcal{X} \subseteq ...
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1answer
283 views

What's the volume of $\{x\in[0,1]^n|\sum x_i\le t\}$ for real $t$?

EDIT: Thanks for your answers and comments. There is indeed a classical easy formula, given by Pietro Majer (with a simple nice proof) in his answer below. Given $x\in\mathbb{R}^n$, $x_i$ denotes ...
42
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2answers
2k views

The view from inside of a mirrored tetrahedron

Suppose you were standing inside a regular tetrahedron $T$ whose internal face surfaces were perfect mirrors. Let's assume $T$'s height is $3{\times}$ yours, so that your eye is roughly at the ...
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119 views

Polytopes with few vertices and few facets

I recently realized that, for fixed $\alpha$ and $\beta$, the number of (combinatorial types of) $d$-polytopes with $\leq d+1+\alpha$ vertices and $\leq d+1+\beta$ facets is bounded by a constant that ...
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1answer
132 views

Regular unimodular triangulation for a certain simplex

Consider an $n$-simplex with vertices given by $(0,0,\dots,r_i,r_{i+1},\dots,r_n)$ where $r_1,\dots,r_n$ are given natural numbers, and $i=0,1,\dots,n+1$. Does this simplex admit a regular, ...
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71 views

What does this number tell me about a convex lattice polygon?

EDIT: I realized I'd tricked myself by working with a too special case of $f$, the question is now updated (boundary lattice points replaced vertices). Suppose I have a convex lattice polygon $P$, ...
5
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2answers
146 views

Positivity of Ehrhart polynomial coefficients

Are there any results stating that a given family of convex polytopes have Ehrhart polynomials with non-negative coefficients? What methods are available for proving such a property for some family ...
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0answers
61 views

Looking for the manuscript “Uniform polytopes” by N. Johnson

The manuscript Uniform Polytopes (1991) by Norman Johnson is cited in the wikipedia page on uniform polytopes (http://en.wikipedia.org/wiki/Uniform_polytope). Is there an electronic copy of this ...
14
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4answers
426 views

The limit of edge-midpoint convex polyhedra

    Starting with a convex polyhedron $P_1 \subset \mathbb{R}^3$, replace that with $P_2$, the convex hull of the midpoints of the edges of $P_1$. Continuing this process, we obtain a ...
7
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2answers
333 views

Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?

Richard Stanley keeps a famous running compilation of different guises of the celebrated Catalan numbers. The number of vertices of the associahedron is one instantiation among the multitude, and the ...
3
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1answer
73 views

Are all marked order polytopes normal?

Richard Stanley showed that order polytopes have a unimodlar ttriangulation. In particular, this implies that they are integrally closed/normal. One can generalize order polytopes to marked order ...
4
votes
2answers
260 views

Convex Sets and Nearest Neighbors

For a set $S \subseteq \mathbb{R}^n$ and a point $x \in \mathbb{R}^n$, let $c_S(x)$ be the point $s \in S$ that minimizes $\|x-s\|$ if such a point exists and is uniquely determined. It is known that ...
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2answers
71 views

Information needed to distinguish combinatorially isomorphic polytopes (up to affine equivalence)

I originally posted this question on Stack Exchange, thinking it perhaps does not qualify as "research-level" but it received no answers... hopefully someone here can help. The title pretty much ...
6
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1answer
187 views

Some polytopes in $\mathbb R^n$ whose vertices have coordinates 1, -1 or 0

Let $n$ and $k$ be positive integers with $k\leq n$. Let $P(n,k)$ be the convex hull in $\mathbb R^n$ of the $2^k {n \choose k}$ vectors whose exactly $k$ coordinates belong to $\{\pm 1\}$ all the ...
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2answers
183 views

Pictures of the von Neumann polytope

Are there any graphic portrayals of von Neumann polytopes in low dimensions?
15
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1answer
237 views

Higher dimensional generalization of: Any quadrilateral tiles the plane?

Any (non-self-intersecting) quadrilateral tiles the plane.     (MathWorld image.) Q. What is the strongest known generalization of this statement to higher dimensions? I.e., ...
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45 views

construction of four dimensional regular convex polytopes

Could anybody give me a reference book about the explicit construction of the 6 regular four dimensional convex polytopes. I cannot easily find Schläfli's original paper so I am looking for a modern ...
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1answer
130 views

Estimating the volume of a union of balls

Let $\{ B_i \}_{i=1}^n$ be a set of $n$ ball in the unit cube $C$ of dimension $d$. If I want to estimate $$ \frac{ \lambda \left( \cup B_i \right) }{\lambda\left( C \right) }, \tag{1} $$ where ...
3
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0answers
54 views

Polynomials connected with Gale's condition and cyclic polytopes

I was reading about cyclic polytopes and (I think naturally) became interested in polynomials $f(X) \in \mathbb{R}[X]$ which have a factorization of the form $f(X)=g(X)g(X+1)$ for some $g(X) \in ...
3
votes
1answer
134 views

$\mathcal{H}$-polyhedron under a linear map

Let $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$ be a (bounded) polyhedron for $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$, $n,m > 0$. Moreover, let $M \colon \mathbb{R}^n \to ...
2
votes
1answer
99 views

Criteria for abstract polytopes to be convex polytope

Suppose I have an abstract polytope defined by a poset. Are there any methods for determining whether the abstract polytope can be geometrically realized as convex-hull on its set of vertices?
2
votes
1answer
87 views

How to (efficiently) find intersection of two polyhedral cones?

I have two polyhedral cones represented by their rays. I am looking to find their intersection, which would also be a polyhedral cone, hopefully efficiently. Does anybody know a way to do that? ...
4
votes
2answers
157 views

Build a topological polytope with a specified CW-structure

I am a topologist and not quite familiar with the tools for building a polytope. I would like to build some topological polytope which is an somewhere in between permutohedron and associahedron which ...
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78 views

Convex Optimization related problem

Suppose two non-negative convex functions $f$ and $g$ be given. We want to solve the following optimization $$\max_{g\leq\epsilon}f.$$ Now suppose that both $f$ and $g$ can be upper-bounded by a ...
5
votes
2answers
217 views

Are shortest halving curves simple closed geodesics?

Let $S$ be a smooth convex surface in $\mathbb{R}^3$ (although my question may as well be asked for the surface of a polyhedron). Say that $\gamma$ is a shortest halving curve if (a) it partitions the ...
2
votes
0answers
73 views

Generators for lattice polytopes

Let $V$ be a finite set of integral vectors in $\mathbb{R}^n$. Consider the real (positive) convex cone $\mathbb{R}_+V$ spanned by $V$. We can assume that all vectors in $V$ are extremal in ...
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2answers
125 views

Volume of normal cone of a simplex (at a vertex)

This question is related to an orthant-type simplex in $\mathbb{R}^n$, which can be defined as $$ S = \{x\in\mathbb{R}_{+}^n: \sum_{i=1}^nx_i \leq 1\}=\overline{\mbox{conv}}(0,e_1,\ldots,e_n). $$ For ...
2
votes
1answer
141 views

intersection of the unit cube and a hyperplane containing the main diagonal

Let $A$ be a linear $m$-dimensional subspace of $\mathbf{R}^n$ $m < n$, containing the point $(1,1,\ldots,1) \in \mathbf{R}^n$, and consider the intersection of $A$ and the unit cube $\Delta_n$ ...
3
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0answers
46 views

Minimax-like theorems involving union and intersection of regions in $\mathbb R^d$

From Minimax theorems, we roughly know that if $f(x,y)$ is convex on $X$ and concave $Y$ for both compact $X,Y$, then: $$ \max_{x\in X}\min_{y\in Y} f(x,y)=\min_{y\in Y}\max_{x\in X} f(x,y). $$ We can ...
4
votes
1answer
135 views

The number of facets of a polyhedron under linear transformation

Consider a (not necessarily bounded) convex polyhedron $P\subset \mathbb{R}^n$ which has $k$ facets. Let $L:\mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation. Question1: Is there a fixed ...
5
votes
1answer
428 views

Linear transformation of a polyhedron

Is there a simple proof that shows: Linear transformation of a $\mathcal{H}$-polyhedron (i.e. the intersection of finitely many closed half-spaces) is a $\mathcal{H}$-polyhedron. Minkowski sum of ...