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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Seeking criteria for “threadable” pairs of centrosymmetric polyhedra

Let $A$ and $B$ be origin-centered centrosymmetric polyhedra in $\mathbb{R}^3$: "for every point $(x, y, z)$ [...] there is an indistinguishable point $(-x, -y, -z)$." Say that $A$ and $B$ are ...
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1k views

Convex hulls of compact sets

Let $A$ be a compact set in a separable Hilbert space $H$, and let $\bar A$ denote its convex hull. Is $\bar A$ compact?
4
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5answers
497 views

Lattice points in dilated polytopes and sumsets

Let $P$ be an integral polytope, that is, the convex hull of some points in $\mathbb{N}^d$. Let $p_1,\dots,p_m$ be all lattice points in $P$. Question: What is the condition on $P$ that guarantees ...
2
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1answer
416 views

Approximation of a convex body by a contained polytope

This question deals with approximating a convex body (a compact convex set of $\mathbb{R}^d$ with non-empty interior) by convex polytopes. For a given $\delta$, let $n_\delta$ be the number of faces ...
4
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1answer
158 views

Toric ideal of slice of a polytope?

Given a collection $A:=\{a_1, \ldots ,a_n \}$ of different integer points in $\mathbb{N}^d$, which span an affine hyperplane when viewed in $\mathbb{R}^d$, one can define a toric ideal $I_A$ from a ...
3
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0answers
114 views

Convex polyhedra jammed in $k$ disjoint holes

For a given convex polyhedron $P \subset \mathbb{R}^3$, I was imagining finding the optimal "fixing" of $P$ in holes (or jamming them in "mud"), which led to the following question. First, scale $P$ ...
6
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1answer
339 views

Has this generalization of a determinant (assigning multiplicities to the rows) been studied?

I'm working on some questions in tropical geometry, and my problem led me to create the following generalization of a determinant: Let $A$ be an $m \times n$ matrix with $m \le n$, and positive ...
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2answers
805 views

An integrality question about expressing an integer as a product of numbers below $n$

Let $n\ge 2$ be a natural number. Suppose that $N$ is a natural number, composed only of primes below $n$, and that can be expressed as $$ N= \prod_{j=1}^{n} j^{x_j} $$ where $x_1$, $\ldots$, $x_n$...
21
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3answers
2k views

How many different numbers can be obtained as product of first $n$ natural numbers?

Let m and n be natural numbers, and consider the set of all possible products of m (not necessarily distinct) elements from the set $\{1,2,\ldots,n\}$, that is consider the set $\{1^{a_1} \cdot 2^{...
6
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1answer
137 views

Maximizing ratio volume/diameter^n by an affinity

Suppose we have a convex compact body $D\subset \mathbb R^n$. We can try to apply affine transformation keeping the volume and decreasing the diameter of $D$. It is clear that there is a constant $\...
2
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0answers
132 views

Definition of self-dual polytope

Given a d-polytope $P$ we define the c-dual polytope as $P^\ast = \{y\in R^d \mid x\cdot y\geq -c, \forall x\in P\}$. Then I say that a polytope is c-polar self-dual if $P=P^\ast$. I cannot find this ...
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0answers
189 views

4D polytope analogues of the icosahedron/Rogers-Ramanujan continued fraction relationship?

The formula for the j-function which employs polynomial invariants of the icosahedron, $$j(\tau)=-\frac{(r^{20} - 228r^{15} + 494r^{10} + 228r^5 + 1)^3}{r^5(r^{10} + 11r^5 - 1)^5}$$ where, $$r^{-1}-...
0
votes
1answer
119 views

2-neighborhood of a simplex

Let $\Delta$ be an $n-1$-simplex in ${\mathbb R}^{n-1}$. For each vertex $v$ of $\Delta$ let $H_v$ be the hyperplane through $v$ and parallel to the opposite facet. By 2-neighborhood of a simplex I ...
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2answers
638 views

“Derived” polyhedra and polytopes

The notion of derived polygon is natural and leads to remarkable convergence. Start with a polygon, and replace it by locating a point on every edge a fraction $\alpha$ between the two endpoints. For ...
6
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0answers
152 views

Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between edges are rational multiples of $\pi$?

After reading these very interesting questions, I came up with another one: Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting ...
10
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1answer
224 views

Why do convex polytope options constrict with dimension, rather than expand?

There are an infinite number of regular polygons in the plane, five regular polyhedra, six regular polytopes in $\mathbb{R}^4$, and then three regular polytopes in every dimension $d > 4$. There ...
7
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1answer
166 views

Convex deltahedra in higher dimensions

There are eight convex polyhedra whose faces are equilateral triangles, so-called deltahedra:        (Image from here) Q. Have the equivalent higher-dimensional ...
15
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166 views

realization spaces of 3-dimensional polytopes

It is a well-know result (Steinitz, 1922) that the realization space of 3-dimensional convex polytopes with fixed combinatorics is contractible. A proof of this theorem can be found for instance in ...
3
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0answers
237 views

Wasserstein distance, convex polytopes and extreme points

Let us consider convex polytopes with $K$ extreme points in $\mathbb{R}^d$. Let $\mathbf{P}$ be such polytope, and let $\text{ext}(\mathbf{P})$ denote its set of extreme points, so that $\mathbf{P} = \...
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0answers
131 views

Distance function from the origin to the boundary of a convex polytope

Let $K\subseteq \mathbb R^n$, $n\ge 3$ be a convex bounded polytope containing the origin in its interior. Consider the unit sphere with geodesic distance $(\mathcal S^{n-1},d)$ and define the ...
3
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1answer
211 views

n-dimensional Delaunay Triangulation of Lattices

I have several questions concerning the Delaunay triangulation of a high dimensional lattice. Given an $n$-dimensional lattice $L$ and its Delaunay triangulation (partition of $R^n$ into simplices ...
2
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3answers
385 views

Repeating an operation infinitely makes any convex $n$-gon a regular $n$-gon?

For any convex $n$-gon $P_{0,1}P_{0,2}\cdots P_{0,n}$, let us consider the following operation : Operation : Let $k=0,1,\cdots$. Take $n$ points $P_{k+1,i}\ (i=1,2,\cdots,n)$ outside of $n$-gon $P_{...
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1answer
229 views

A construction related to scissors congruence

I was thinking about the following some time ago. My question is whether such things have been studied before. Let $E_n$ be the abelian group with a generator for each (bounded) euclidean polytope of ...
6
votes
2answers
656 views

“MultiCatalan numbers”

Could anyone provide a reference for the following (sort of) generalization of Catalan numbers: the multinomial coefficient $$ \binom{2k_1+3k_2+4k_3+...}{k_1+2k_2+3k_3+...,k_1,k_2,k_3,...} $$ is ...
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vote
2answers
419 views

Vertices of a Polytope

Given two convex polytopes $P,Q$ such that $P\subset Q$. We are given that all the vertices of $P$ are also vertices of $Q$ and all the facets defining planes of $Q$ are also facets defining planes of ...
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2answers
6k views

The amplituhedron minus the physics

Is it possible to appreciate the geometric/polytopal properties of the amplituhedron without delving into the physics that gave rise to it? All the descriptions I've so far encountered assume ...
4
votes
2answers
262 views

point in polytope

Suppose I have the convex hull $P$ of a finite collection of points in $\mathbb{R}^d,$ and I want to see whether a point $p$ is contained in $P.$ This is a standard (some would say the standard linear ...
11
votes
2answers
860 views

Efficiently determine if convex hull contains the unit ball

Given a set of $n$ points in $\mathbb{R}^d$, is there an algorithm to determine if the convex hull contains the unit ball centered at the origin in polynomial time? The convex hull itself might have ...
2
votes
1answer
112 views

Estimates on the number of vertices of reflexive polytopes

Suppose $M \cong \mathbb{Z}^n$ is a rank $n$ lattice, with dual lattice $N$. Suppose $\Delta$ is a full dimensional lattice polytope (i.e. convex hull of finite lattice points) in $M$. Then $\Delta$ ...
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228 views

Right-angled polytopes

%This question is motivated by the little discussion here at the bottom. The following thing are known about hyperbolic right-angled polytopes: Compact hyperbolic right-angled polytopes do not ...
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vote
1answer
107 views

Submodular measures on the hypercube

By the hypercube I mean the lattice formed by all n-bit strings ordered by pointwise inequality. For example, $000 \leq 110$, $010 \leq 110$, $110$ and $001$ are not comparable. Further we have the ...
5
votes
2answers
170 views

name for a polytope constructed from a system of linear equations?

To a system of inhomogeneous linear equations, one can associate a polytope, as follows. Let $A\in\mathbb{R}^{m\times n}$, $b\in\mathbb{R}^m$ and $V=\{x\mid Ax=b, \text{support of $x$ minimal}\}\...
4
votes
1answer
238 views

n-simplex in an intersection of n balls

Consider any $n$-simplex, $n \geq 2$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ in the simplex. The question: ...
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0answers
236 views

Bi-spherical polyhedra

Bicentric polygons have been studied: a polygon all of whose vertices lie on its circumcirle, and whose incircle is tangent to every edge:   I have not been able to find a comparable literature ...
14
votes
1answer
312 views

Smallest regular simplex containing the unit cube in $R^n$

What is the length $e_n$ of the edge of the smallest $n$-dimensional regular simplex $S_n$ containing the $n$-dimensional unit cube $Q_n$? In particular, is there $n$ such that $e_n<\sqrt{2}(n+...
5
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0answers
176 views

Tetrahedra passing through a hole

Assume a plane $P\subset\mathbb R^3$ has a hole $H$, and that the hole is topologically a compact disc. Being so, $P\setminus H$ does not separate the space. A regular tetrahedron $\sigma^3$ (of edge-...
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2answers
253 views

Hales's fan associated with a polyhedron

In Hales's book (cited below), he associates what he calls a fan with any convex polyhedron in $\mathbb{R}^3$. I will not define his notion of fan, but let his figure (p.137) serve as a definition: &...
8
votes
3answers
843 views

On maximal regular polyhedra inscribed in a regular polyhedron

Let T, C, O, D, or I be regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron, respectively. Suppose that the outer polyhedron have edge-length 1. For example, it's easy to prove that ...
10
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2answers
234 views

Do maximal polyhedra have algebraic volume?

Is it possible to prove that for every $n > 3$ the maximal possible volume of a convex polyhedron having $n$ vertices inscribed in a sphere of unit radius is an algebraic number? Update: What ...
2
votes
1answer
168 views

Equiprojective polyhedra

Seeing Garabed Gulbenkian's question (which was inspired by Joel Hamkins' question), reminds me of an analogous problem which I believe remains open, and which some might find intriguing. Define an ...
5
votes
2answers
343 views

Covering convex polygons with inscribed disks

The following problem came up when discussing mapping software (e.g., Google maps) with computer scientists. By $B(c,r)$ I mean the planar disk (open or closed, it doesn't matter) of radius $r$ around ...
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3answers
240 views

Cellular decomposition of a sphere to polytope

My primary question is: given a cellular decomposition of a sphere is there any way to check if it can be embedded as the boundary of a polytope? My question is motivated by the following problem. I ...
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0answers
143 views

existence of lattice point in polytope

This question was probably asked before but here goes. I have a convex polytope given by $Ax\leq b$ for a specific integer matrix $A$ and integer vector $b$. I need a simple method/result on how to ...
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0answers
74 views

Optimization over a variable domain defined as a convex hull of given points [closed]

I have an optimization problem: $\max_{\bf{x}} Z(\bf{x})$, s.t. $\bf{x} \in conv(\bf{S})$ where $\bf{x}$ is an $n$-dimensional vector, $Z(\bf{x})$ is a non-linear function. The domain of $\bf{x}$ ...
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2answers
242 views

Volumes of convex vs non-convex polyhedra with prescribed facets areas

It is known that given a set of Areas $A_f$ and normals $\vec{n}_f$ if $\sum_f A_f \vec{n}_f=0$ exist a unique convex polyhedron with given face areas and normals. (Minkowski theorem - See Alexandrov ...
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3answers
239 views

How to determine if two rational cones intersect?

Let $$\displaystyle C_1=C(r_1,...,r_{n_1})=( x\in \mathbb{R^d}|x=\sum_{i=1}^{n_1}\lambda_i r_i, \lambda_i\in \mathbb{R_{>0}})$$ $$\displaystyle C_2=C(t_1,...,t_{n_2})=(x\in \mathbb{R^d}|x=\sum_{i=...
3
votes
1answer
275 views

Lattice points in cross-polytopes

Let $E\subset \mathbb{R}^n$ be a cross-polytope: $$E= \left\lbrace x : \frac{|x_1|}{q_1}+\cdots+\frac{|x_n|}{q_n}\leq 1 \right\rbrace, $$ where $q_1,\dots,q_n$ are positive integers. I am interested ...
7
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1answer
180 views

Do there exist “expanding” $1$-skeletons of simple $4$-polytopes?

Let $\{ G_n \}_{n \ge 1}$ be a sequence of graphs such that the number of vertices of $G_n$ tends to $\infty$ as $n \to \infty$. We say that $\{ G_n \}_{n \ge 1}$ is an expander family if $\lambda_2( ...
4
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1answer
196 views

Zoll Flat Finsler tori and convex bodies on a starry night

The starry night. The "celestial sphere" is given by set of non-zero vectors in $\mathbb{R}^n$ modulo positive dilations (i.e., $v \equiv w$ if $v = \lambda w$ for some $ \lambda > 0$) and the "...
4
votes
1answer
249 views

convex polyhedron in the unit cube

Let $P$ be a given finite set of points within the $n$-dimensional unit cube. A finite set $Q$ of points within the $n$-dimensional unit cube covers $P$ if $\operatorname{conv}(Q) \supseteq P$ where $\...