**10**

votes

**2**answers

787 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 $x_1$, $\ldots$, ...

**21**

votes

**3**answers

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 ...

**6**

votes

**1**answer

136 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**

votes

**0**answers

129 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 ...

**5**

votes

**0**answers

183 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,
...

**0**

votes

**1**answer

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 ...

**16**

votes

**2**answers

592 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**

votes

**0**answers

143 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**

votes

**1**answer

222 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**

votes

**1**answer

165 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 ...

**13**

votes

**0**answers

159 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**

votes

**0**answers

232 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} = ...

**1**

vote

**0**answers

126 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**

votes

**1**answer

207 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**

votes

**3**answers

379 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 ...

**4**

votes

**1**answer

227 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

**2**answers

646 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 ...

**1**

vote

**2**answers

416 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 ...

**54**

votes

**2**answers

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

**2**answers

247 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

**2**answers

820 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

**1**answer

110 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 $\Delta$ ...

**10**

votes

**0**answers

227 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 ...

**1**

vote

**1**answer

106 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

**2**answers

165 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 $x$ ...

**4**

votes

**1**answer

232 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: ...

**9**

votes

**0**answers

234 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

**1**answer

303 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 ...

**4**

votes

**0**answers

164 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 ...

**1**

vote

**2**answers

244 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

**3**answers

832 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**

votes

**2**answers

231 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

**1**answer

166 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

**2**answers

336 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 ...

**5**

votes

**3**answers

236 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 ...

**2**

votes

**0**answers

140 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 ...

**2**

votes

**0**answers

70 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 $\bf{x}$ ...

**7**

votes

**2**answers

235 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 ...

**1**

vote

**3**answers

227 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 ...

**3**

votes

**1**answer

271 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**

votes

**1**answer

168 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**

votes

**1**answer

193 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

**1**answer

243 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 ...

**10**

votes

**1**answer

325 views

### Needle probing for a convex body

Suppose there is an unknown closed convex body $K$ of
volume vol$(K) = V$ inside the
unit cube $[-\frac{1}{2}, \frac{1}{2}]^d$ in $\mathbb{R}^d$.
You are permitted to probe with a (one-dimensional)
...

**8**

votes

**3**answers

504 views

### Rate of growth of an explicit integral

Let $$J_1=\int_0^1\frac{1}{\sqrt{1-t_2}}dt_2,$$
$$J_2=\int_0^1 \int_0^{t_2}\frac{1}{\sqrt{1-t_2}}(\frac{1}{\sqrt{1-t_3}}+\frac{1}{\sqrt{t_2-t_3}})dt_3dt_2,$$
$J_3=\int_0^1 ...

**9**

votes

**1**answer

412 views

### Question about tetrahedron decomposition

Are there tetrahedra which can be subdivided into three non-overlapping parts similar to the original? I believe this would require splitting one face into three parts. I know some types of tetrahedra ...

**10**

votes

**1**answer

472 views

### When is a matrix similar to a non-negative matrix?

Consider a real square matrix $A$ of size $n\times n$. Under which conditions on $A$ does there exist a row-stochastic matrix $U$ (non-negative, rowsums = 1), such that $A'=U^{-1}AU$ is a non-negative ...

**3**

votes

**1**answer

173 views

### Simplex with edges of length at least s having smallest circumradius

Is it true that of all $n$-simplices with edge lengths greater than or equal to some parameter $s$, the regular simplex with edge lengths $s$ has the smallest circumradius? It seems obvious, but I ...

**5**

votes

**2**answers

251 views

### Diameter-area ratio for affine tranformations.

Consider an convex plane figure $F$. How to prove that there is an affine transformation $a$ such that $\sqrt{3}$ diameter$(a(F))^2\leq 4$ area$(a(F))$?
I found only one reference, to "Über einige ...

**0**

votes

**2**answers

269 views

### Higher dimensional convex hull

Let $CH(S)$ be a convex hull of a finite set $S$ and denote the set of all the vertices of $CH(S)$ as $Vert(S)$. For a vertex $v \in Vert(S)$, it has an associated set $E(v)$ which is defined as ...