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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Non-negative polynomials $f(p), p\in P$ from Polynomial ideal where $P$ compact polytope?

Partially related to Non-negative Polynomials from Polynomial Ideal? where focus on graph theory but now I want to understand more generally how to determine non-negativity in more general case. A. ...
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estimating binomial coefficients

There is a beautiful paper on the arXiv by Andrew Suk containing an asymptotic result about the Erdös-Szekeres convex polygon problem. I am struggling with one of the estimates he makes on page 4. He ...
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Convex polytopes as “products” of lower dimensional polytopes of the same family

This MO answer on enumerative geometry details the sense in which an associahedron is a product of lower dimensional associahedra, and the comments in this MSE-Q indicate the same is true for ...
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Is the chromatic number of the graph of the permutahedron known?

The permutahedron $\Pi_n$ is the polytope that is the convex hull of all permutations of the vector $(1,2,...,n)$. There are many results on its structure, but I couldn't find a result on the ...
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Inequalities for marginals of distribution on hyperplane

Let $H = \{ (a,b,c) \in \mathbb{Z}_{\geq 0}^3 : a+b+c=n \}$. If we have a probability distribution on $H$, we can take its marginals onto the $a$, $b$ and $c$ variables and obtain three probability ...
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Lattice points in regular simplex

Suppose we are given a regular (closed) simplex $S$ in a vector space $V$ of dimension $n$, whose vertices have integer values. Then for a lattice $L$, is there a sufficient criterion, for $S$ to ...
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Terminology and technique for repeated pairwise removal of elements of posets: “Collapsibility” of a “face poset”

Let $P$ be a poset, or partially ordered set. Let $\le$ denote the reflexive order on $P$, and $<$ the corresponding irreflexive order. Let the phrase "a maximal pair" in $P$ refer to an ordered ...
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Relating face polytopes of permutohedra to integer partitions

The OEIS entries A019538, A049019, and A133314, relate a refinement of the face polynomials of the permutohedra (A049019) to partition polynomials (A133314) defined by multiplicative inversion of an ...
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134 views

Poincare-Hopf theorem for polytopes?

Is there an analogue of Poincare-Hopf theorem for polytopes? I want to apply it in the following situation. I have a polytope in $R^n$ and a smooth explicitly given vector field in $R^n$. I want to ...
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First to note/document the relation between permutohedra and multiplicative inversion

The relation between the refined face numbers of the permutohedra and the formal series expansion of the reciprocal of a function (exponential generating function, formal Taylor series) is given in ...
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Rational $d$-simplices

Define a rational $d$-simplex as a simplex in $\mathbb{R}^d$ such that the measure of all its $k$-dimensional faces, $k \ge 1$, is rational. So a rational triangle has rational edge lengths and ...
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Complete the following sequence: point, triangle, octahedron, . . . in a dg-category

Let $\mathcal C$ be a pre-triangulated dg-category (or a stable $\infty$-category, if you wish). An object $X$ in $\mathcal C$ gives a "point": $$X$$ A morphism $X\xrightarrow f Y$ in $\mathcal C$ ...
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Average nastiness of a Newton polytope

Given a Newton polytope $P$ inside the $d$-scaled simplex ( i.e $\alpha \in P \implies | \alpha |=d $ ), then we define the following quantity: $$ P(x)= \left( \sum_{\alpha \in P} \binom{d}{\alpha}...
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Zeros of Hilbert series of affine toric varieties

Consider a convex rational polyhedral cone $C\subset\mathbb R^m$ with vertex at the origin. Let $X$ be the corresponding affine toric variety, i.e. $\mathbb C[X]=\mathbb C[\mathbb Z^m\cap C^\circ]$. ...
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1answer
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Counting faces on multipermutahedra/multipermutohedra

A multipermutahedron is the convex hull of all permutations of a list of numbers. For example, $\Pi(0,1,2)$ generates a regular hexagon, and $\Pi(0,1,1,2)$ generates a cuboctahedron. In general, ...
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Tensor Product of Convex Sets?

I was wondering if such a concept was used anywhere. What I was thinking of is this. Consider two vectors spaces $V,W$ and convex sets $C_1 \subseteq V$ and $C_2 \subseteq W$ if we define $C_1 \otimes ...
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Rational inscribed realization of the regular dodecahedron

While it is clear that the regular dodecahedron $D$ cannot be realized with all integer coordinates, it is easy to find a polytope, which is combinatorially equivalent (face lattice isomorphic) to $D$ ...
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regular triangulations of the product of two simplices

Is description of all regular triangluations of $\Delta^n\times \Delta^k$ known? (Regular triangulations are those which correspond to vertices of Gelfand--Kapranov--Zelevinsky secondary polytope, or, ...
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Distinguishing (possibly lower dimensional) $1$-skeleton of a regular graph inscribed in a sphere

Consider you have two (possibly same) convex $1$-skeleton of a regular graph $A$ and $B$ in $m$-dimensions inscribed in a sphere with possibly exponential number of vertices in $n$-dimension with ...
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Partitioning a convex object without harming existing convex subsets

$C$ is a convex planar figure and $C_1,\dots,C_n$ are pairwise-disjoint convex subsets of $C$, like this: A convex-preserving partition of $C$ is a partition $C=E_1\cup\dots\cup E_N$, , such that $...
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Is every triangulation of a Euclidean ball by convex tetrahedra shellable?

Suppose you are given a 3-ball $B$ in $\mathbb{R}^3$ that is bounded by a PL sphere, a triangulation $T$ of $B$ by Euclidean tetrahedra. Is that triangulation necessarily shellable? I know that if $...
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247 views

Covering the unit sphere by open hemispheres

Suppose $H_1,\ldots,H_{2n}$ are open hemispheres which cover $S^{n-1}$ with the property that removing any one of them leaves $S^{n-1}$ uncovered. Is it necessarily the case that the hemispheres can ...
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Quick way to compute Ehrhart polynomial of Young diagram posets?

Using the hook formula, it is easy to compute the volume of order polytopes obtained from posets with partition shape, since this is the same as the number of linear extensions. To my knowledge, ...
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What are the 4 convex simplicial 4-polytopes that have 6 vertices?

In Convex polytopes and related complexes by Klee and Kleinschmidt they state the number of $d$-polytopes with $d+2$ vertices is $\lfloor \frac{d^2}{4}\rfloor$. I was wondering what the four $4$-...
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Positivity of coefficients of Ehrhart polynomial of n-Tetrahedron

A set of positive integers $d_1, \dots, d_n$ describe two n-dimensional closed lattice tetrahedron: $$ T = \left\{ (x_1, \dots, x_n) \in \mathbb{R}^n: \sum_{i=1}^n \frac{x_i}{d_i} \leq 1 \textrm{ and ...
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Is the complement of a vertex figure in an (abstract) polytope connected?

I consider an (abstract) regular polytope $P$, and $H$ a vertex figure of $P$. Is the complement $P \setminus H$ connected (as a poset, in the sense that its Hasse diagram, ignoring the improper faces,...
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1answer
66 views

Efficiently Generating the Convex Hulls of Two Polytopes and Counting Faces

Suppose you have two polytopes $P_1, P_2 \in \Bbb{R}^n$ given by $$ P_1 = \lbrace x: A_1 x \le b_1\rbrace$$ $$ P_2 = \lbrace x: A_2 x \le b_2\rbrace $$ I wish to find their convex hull, that is a ...
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216 views

Basic question about polytope duals

The following must be well known. Is there a beginning or midlevel text where the answer is discussed? Thanks. Along with a polytope one has the notion of its dual which is officially defined via ...
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151 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: ...
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1answer
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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 ...
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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 ...
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1answer
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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 (...
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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 $P$ (...
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3answers
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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 ...
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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
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2answers
120 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 ...
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73 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. ...
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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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37 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 ...
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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 $p_1<\ldots<...
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51 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 ...
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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 ...
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63 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) := \{(i,j)...
6
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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 ...
4
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1answer
130 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 ...
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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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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} ...
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1answer
137 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, ...
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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 ...
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47 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 $n$...