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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258 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, ...
2
votes
2answers
103 views

Number of facets of a polyhedron when a vertex is removed

My question, informally: I have a bounded polyhedron in R^n with k facets, and I want to remove a vertex of this problem. How many facets does the remaining polyhedron have at most? More formally: ...
1
vote
2answers
40 views

Can we prove that a normal surface of an extreme point of a convex subset of a simplex is a separating hyperplane?

Let's assume that we have a simplex $G = \{x\in R^d|\sum_{i=1}^d x_i=1, x_i\ge 0 , i = 1, 2, .., d\}$ and a polyhedral convex subset $H \subseteq G$. Is it possible to prove that for any extremal ...
4
votes
1answer
279 views

Simplex in convex polytope, pulling triangulation

Let $P$ be a convex $d$-dimensional polytope. I have two questions, related to triangulations of $P$. Question 1: Let $p$ be in the interior of $P$. Can I always find a triangulation of $P$, such ...
7
votes
1answer
120 views

Classifying two-faces of four-polytopes

Motivation: This question is related to my study of hyperbolic Coxeter polytopes. In general, if one put some restrictions on the type of their dihedral angles (say, all dihedral angles are equal to ...
2
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0answers
54 views

Approximating Ehrhart Polynomial of Rational n-Tetrahedron

A set of positive integers $d_1, \dots, d_n$ describe a n-dimensional tetrahedron $T$ with the vertices $$ (0,\dots,0), (1/d_1,0,\dots,0), (0, 1/d_2,\dots,0), \dots, (0,\dots,1/d_n).$$ Let $L_T(t)$ be ...
6
votes
2answers
149 views

Gaussian and the convex hull of moment curves

Let $c_1,\dots, c_d$ be the first $d$ moments of the standard normal distribution. Does the point $(c_1,\dots, c_d)$ lie in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$, for a ...
8
votes
1answer
189 views

Tighter Caratheodory on the moment curve?

The moment curve is the set of points of the form $$(t,t^2,t^3,...,t^n) \in R^n$$ Let $M$ be the portion of the moment curve where $t\in [0,1]$, and let $\overline{M}$ be the convex hull of $M$. ...
6
votes
2answers
218 views

“Database” of simplicial polytopes/spheres

Reading through various papers on polytopes I have come across really interesting examples of simplical polytopes and non-shellable (or non-PL) simplicial spheres but sometimes it is hard to keep ...
5
votes
1answer
63 views

Sample integer points of cross-polytope uniformly

For $r,d\in\mathbb{N}$, let $$C_{r,d}=\{x\in\mathbb{Z}^d: \|x\|_1\le r\}\subset\mathbb{Z}^d$$ be the set of integer points of the $d$-dimensional cross-polytope with radius $r$. What is ...
4
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0answers
112 views

Reference for the notion of polyhedra “degenerations”

Let $P$ be a convex polyhedron and let $P(t)$ be a continuous deformation thereof, such that: a) $P(0)=P$; b) for all $t\in[0;1)$ the polyhedron $P(t)$ is strongly combinatorially equivalent to $P$ ...
1
vote
2answers
101 views

An optimality condition for the corners of convex polytopes?

Let $H,H'\subset\mathbb{R}^m$ be two hyperplanes with unit normal-vector, and let $P\subset\mathbb{R^m}$ be a convex polytope (defined via its corners $v_0, ... , v_n$, where $n\ge m$). Let's ...
28
votes
2answers
1k views

The logic of convex sets

Let me start with Helly's theorem: Let $A_1$, $A_2$, ..., $A_{n+2}$ be $n+2$ convex subsets of $\mathbb R^n$. If any $n+1$ of these subsets intersect (this means: have nonempty intersection), the so ...
4
votes
2answers
100 views

Cluster Variables for non-convex n-gons

Most of the lectures and lecture notes on Cluster Algebras (at least from Combinatorial point of view) start with mutations of the diagonals of a convex n-gon (mostly the pentagon) as the illustration ...
3
votes
1answer
119 views

The center of a minimal convex superbody

Is the following true? CONJECTURE: $\,$ Let $\ B\ C\subseteq\mathbb R^n\ $ be convex bodies in $\mathbb R^n$ such that $\ C\ $ is centrally symmetric, $\ B\subseteq C,\ $ and $\ t\!\cdot\! B\ $ ...
3
votes
2answers
105 views

How many different integer polytopes does square lattice have?

Let $E_n = \{ (i,j) : 0 \leq i,j \leq n-1 \}$. We say that a polytope $P$ is an integer polytope in $E_n$ if all vertices of $P$ belongs to $E_n$. My question is how many different integer polytopes ...
1
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0answers
38 views

Projecting on a convex compact polytope with special form

Let $E$ be a large sparse $l$-by-$n$ matrix ($l$ and $n$ can be in the billions...) with coefficients in $\{-1, 0, 1\}$: the first row of $E$ is the vector $(1,0,0,\ldots,0) \in \mathbb R^n$, and ...
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0answers
68 views

Projecting on a a special polyhedron

Let $X$ be an $n$-by-$p$ matrix and consider the closed convex polyhedron $$\mathcal P_X := \{y \in \mathbb R^n | \|X^Ty\|_\infty \le 1\}.$$ Notice that $\mathcal P_X$ is symmetric about the origin. ...
1
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0answers
36 views

Checking integer points of given infinity norm on intersection

If we have convex region $\mathscr R$ given as intersection of a convex polytope $\mathscr P$ and an ellisoid $\mathscr E$ in $\ell^2$ norm is there an efficient way to test if there is an integer ...
4
votes
0answers
75 views

Face structures of chain polytopes

For a finite poset $P$ the chain polytope $\mathscr C(P)\subset\mathbb{R}^P$ consists of such $g$ that $g(p)\ge 0$ for all $p\in P$ and $$g(p_1)+\ldots+g(p_n)\le 1$$ for any chain ...
6
votes
2answers
192 views

Which unfoldings of the hypercube tile 3-space: How to check for isometric space-fillers?

Recently Mark McClure constructed and displayed the 261 unfoldings of the hypercube (tesseract) in response to the question, "3D models of the unfoldings of the hypercube?": The first 9 unfoldings ...
3
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0answers
44 views

Delzant polytopes and combinatorial types

At first, let us see the following matheoverflow question, About a Delzant polytope. (In particular dodecahedron) The question is whether (combinatorial) regular dodecahedron can be realized as a ...
4
votes
2answers
55 views

Affine hull of a set of non-negative matrices with fixed row-sums

Fix any non-negative matrix $M \in \mathbb{R}_{\geq 0}^{m \times n}$ that contains no zero-row and no zero-column. Further, fix any positive vector $r \in \mathbb{R}_{> 0}^m$. With $nz(M) := ...
4
votes
1answer
120 views

Coefficients of Ehrhart polynomials, in the binomial-coefficient basis

Let $P$ be the convex hull of a finite set of points in $\mathbb Z^d$, and $p(n) = \#\{nP \cap \mathbb Z^d\}$ be its Ehrhart polynomial, which is also the Hilbert polynomial of the corresponding ...
5
votes
0answers
51 views

Convex hull of the orbit of a matrix under permutations

Let $P$ be a generic permutation matrix on $\mathbb{R}^n$. For any vector $x \in \mathbb{R}^n$, the convex hull of the set $\{ Px : \; \text{$P$ is a permutation matrix}\}$ is the set of vectors ...
5
votes
1answer
131 views

Are the primary parallelotopes classified? (equivalently, Voronoi cells of lattices)

A primary parallelohedron is a polyhedron that can fill space with infinite translated copies. It is known (e.g., Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 29-30, 1973; or, ...
11
votes
0answers
152 views

Is combinatorial automorphism of symmetric convex polytope always odd?

The question is formulated in the title. More precisely, if $P$ is an origin-symmetric convex polytope in $\mathbb{R}^d$, and $f$ is a bijective transform of the set of the vertices of $P$, which ...
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votes
0answers
23 views

Efficient sampling from a polytope with large number of contraints [duplicate]

As far as I know, the most popular way to sample from a polytope (in H-representation) \begin{equation} \mathcal{P} := \{z \in \mathbb{R}^n | (Az)_j \le b_j\; \forall j=1,2,\ldots,m\} \end{equation} ...
17
votes
2answers
778 views

About a Delzant polytope. (In particular dodecahedron)

Hi. I have a question. Definition. Delzant polytope $P$ is a rational convex simple polytope with the smooth condition. Here, "smooth" means that for each vertex $v$, the $n$ edges containing $v$ ...
1
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0answers
42 views

Number of simplices contained in a convex body

I am interested in the following question: Given a convex body $K$ in $\mathbb{R}^d$ and an $\epsilon>0$ small enough, how many $d$-simplices $\{D_i\}_{i=1}^m\subset K$ do we need at least so that ...
5
votes
1answer
758 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 ...
1
vote
0answers
41 views

Volume of intersection of a convex polytope with general affine space

This question generalizes (this question) on the same site. Let $\Delta^{n}$ denote the $n$-dimensional simplex in $n$ dimensions. That is, $\Delta^{n}$ is the convex closure of the origin and the ...
7
votes
7answers
2k views

Uniformly Sampling from Convex Polytopes

How to choose a point uniformly from a convex polytope $P \subset [0,1]^n$ defined by some inequalities, Ax < b ? (Here A is an m-by-n matrix, x is n-by-1 and b is m-by-1.) I imagine that you ...
11
votes
2answers
806 views

Approximating a convex function by a piecewise linear function

Suppose I have a Lipschitz-continuous convex function $f:\mathbb{R}^n\rightarrow \mathbb{R}$. I wish to approximate it on the unit ball by a piecewise-linear function $g:\mathbb{R}^n\rightarrow ...
2
votes
1answer
73 views

Internal edges in Convex Polytopes

Suppose $S\subset{\mathbb R}^n$ is an infinite subset that is in general position which means that the intersection of $S$ with every affine subspace of dimension $d<n$ always contains at most ...
7
votes
2answers
138 views

$f$-vector of simple convex polytope via directions of facets

Let $P$ be a simple convex polytope in $\mathbb{R}^d$ (that is, any vertex belongs to exactly $d+1$ facets). Given the collection of outer normals to facets of $P$, combinatorics of $P$ may be ...
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votes
0answers
42 views

log convexity for the norm of vector-valued function

Log convexity of various functions defined on the space of Hermitian matrices plays an important role in matrix analysis and probability theory. Given $v \in \mathbb{C}^n$, $D$ a diagonal matrix with ...
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0answers
44 views

Inscribed polytopal approximation to a convex body

This question is on the continuation of the post Approximation of convex body by polytopes The central problem I am interested is an explicit construction of inscribed polytope with at most $n$ ...
6
votes
1answer
173 views

What polytope is this? Bounded sums with choice of coefficients

Let $b\gt a\gt 0$ be constants. Define $P_n(a,b)$ to be the set of all $(x_1,\ldots,x_n)\in\mathbb{R}^n$ satisfying $$ |c_1 x_1 + \cdots + c_n x_n| \le 1$$ for every choice of ...
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35 views

Under which conditions is the union of conic hulls of sets in a cartesian product equal to $\mathbb{R}^N$?

Question: Under which conditions on $A, B\in\mathbb{R}^{N\times N}$ is the function $f: \mathbb{R}^N\mapsto \mathbb{R}^N$, $$f(v) = A[v]_+ + B[-v]_+$$ surjective? Here $[.]_+$ is an elementwise ...
0
votes
1answer
40 views

Mapping n variables to the surface of an n+1-dimensional cross polytope [closed]

I'm looking for a function that maps n variables to points on the surface of an n+1-dimensional cross polytope. For example, given one variable, the function would return a point on the perimeter of ...
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votes
2answers
90 views

Maximal-Orthogonal Convex Hull (or Maximal-Rectilinear Convex Hull) [closed]

Edit : Consider giving a reason for down vote. In my research, I have come across a this paper from the Computational Geometry field and I am not able to understand the concept of ...
2
votes
0answers
73 views

Testing if a point is inside a convex polytope formed by halfspaces in n-dimension

Assume we have a convex polytope that is formed by the intersection of $k$-halfspaces in $\mathbb{R}^{n}$. $$ a_{0,0}x^{n-1} + {a}_{0,1}x^{n-2} + ... a_{0,n-1} \leq 0 $$ $$ a_{1,0}x^{n-1} + ...
2
votes
0answers
70 views

About optimizing a convex function on a hypercube

Given a real valued convex function $g$ on $[-1,1]^n$, let $f$ be the restriction of it on the hypercube $\{-1,1\}^n$. I want to find a vertex on the hypercube $\{-1,1\}^n$ on which either (1) $f$ ...
3
votes
0answers
35 views

Is every 1-skeleton of a 4-tope Steinitzian?

Call a graph "polyhedral" if it is simple, planar, and 3-connected. For example, the 1-dimensional skeleton of every 3-dimensional convex polytope, regarded as a graph, is polyhedral. By a theoreom ...
4
votes
1answer
102 views

Are all marked order polytopes normal?

Richard Stanley showed that order polytopes have a unimoudlar triangulation. In particular, this implies that they are integrally closed/normal. One can generalize order polytopes to marked order ...
3
votes
0answers
36 views

“Spherelike” embedding for smooth Fano polytope

Let $\mathbb{R}^n$ be a vector space with Euclidean inner product and $\mathbb{Z}^n \subset \mathbb{R}^n$ be a lattice. Let $P$ be a full dimension convex lattice polytope in $\mathbb{R}^n$ containing ...
5
votes
1answer
71 views

Volume satisfying inequality constraints (simplex subset)

Is there a way to find the volume of the "feasible region" of a standard simplex satisfying simple range constraints? $x_1+x_2+...+x_n = 1$ $a_1 \le x_1 \le b_1$ $a_2 \le x_2 \le b_2$ $...$ $a_n \le ...
1
vote
1answer
71 views

approximate diameter of polytopes in high dimensions

I just came across the following problem: Let us consider the unit corner of the n-cube $$ \Delta^n = \left\{(t_1,\cdots,t_n)\in\mathbb{R}^n\mid\sum_{i = 1}^{n}{t_i} \leq 1 \mbox{ and } t_i \ge 0 ...
1
vote
1answer
79 views

4D Duoprisms based on nonconvex polygons

A duoprism is a polytope that can be expressed as the Cartesian product of two polytopes (each of dimension $\ge 2$). Four-dimensional duoprisms in particular have been studied: $$P \times Q = \{ ...