**5**

votes

**6**answers

2k views

### coordinates of vertices of regular simplex

For $d=3$, vertex coordinates of a regular simplex have a simple expression since vertices correspond to four vertices of a cube. Is there a simple expression for higher dimensions? In particular I'm ...

**1**

vote

**1**answer

189 views

### Cardinality of non-integer points in the translation of the Minkowski sum of convex hull.

Let $\operatorname{conv}(a_1,\ldots,a_m)$ denote the convex hull of $\{a_1,\ldots,a_m\}$. Let $\mathbb{Z}_+=\mathbb{N}\cup\{0\}$ and $\mathbb{Q}_+$ denotes the positive (inluding 0) rational numbers. ...

**0**

votes

**0**answers

24 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 $n$ ...

**3**

votes

**1**answer

104 views

### Find the minimum distance between two convex hulls

We work over $\mathbb{R}^N$. Let $\mathbf{P}_1$ denote the hyperplane constructed using $N$ points, each of which is on a different axis (there are $N$ axes). We denote by $\mathbf{P}_2$ the convex ...

**5**

votes

**0**answers

39 views

### Measure-minimizing simplex with fixed inradius

Let $\Delta^n$ be an $n$-simplex in $\mathbb{R}^n$. Let $V$ be the volume and $r$ the inradius (radius of the inscribed sphere) of $\Delta^n$. There is a well-known result that
$$
V \geq ...

**1**

vote

**1**answer

185 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, ...

**0**

votes

**0**answers

17 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 $n\ge m$).
lets further ...

**0**

votes

**0**answers

43 views

### Finding a movement taking out of a convex set

There is a convex set $S$ as the hull of M points in an D-dimensional Euclidean space and a point $\vec P$ in the set. Then, there is a set of vectors $\vec W$ taking the form $\vec W=\sum_{i=1}^N ...

**6**

votes

**4**answers

90 views

### Uniform Sampling Subject to Linear Equalities and Non-Negativity Constraint

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints
$$A\vec{w}=\vec{b} \ \ and \ \ \vec{w} \geq ...

**4**

votes

**1**answer

89 views

### How to prove the existence of the polytope in $\mathbb{R}^d$ with a given number of faces, minimizing the isoperimetric ratio?

This is the isoperimetric type question. We know that in $\mathbb{R}^d$, balls are the sets that minimize the isoperimetric ratio $\frac{S^{d}}{V^{d-1}}$, where $S$ is the surface area and $V$ is the ...

**4**

votes

**0**answers

87 views

### Prove or disprove a claim about covering a polytope by convex polytopes in a certain way

Here is the claim:
Given a polytope $K$ in a unit ball in $\mathbb{R}^d$, there exists a universal constant $C(d)>0$ depending only on $d$ and a countable collection of convex polytopes ...

**17**

votes

**1**answer

478 views

### Egalitarian measures

A question I got asked I while ago:
If $T$ is a triangle in $\mathbb R^2$, is there a function $f:T\to\mathbb R$ such that the integral of $f$ over each straight segment connecting two points in the ...

**1**

vote

**0**answers

60 views

### About properties of polynomials with common interlacing

Say $\{a_1,a_2,..,a_n\}$ and $\{b_1,b_2,...,b_n\}$ be the real roots of two monic polynomials of degree $n$ which have a common interlacing. (say I have arranged the roots in increasing order)
Can ...

**20**

votes

**10**answers

9k views

### Algorithm for finding the volume of a convex polytope

It's easy to find the area of a convex polygon by division into triangles, but what is the optimal way of finding the volume of higher-dimensional convex bodies? I tried a few methods for dividing ...

**7**

votes

**0**answers

114 views

### How many facets can $\{\|D^T x\|_1\leq 1\}$ have?

$\newcommand{\RR}{\mathbb{R}}$Consider $x\in\RR^n$ and $D\in \RR^{n\times p}$ with $p\geq n$ and full rank. My question is:
How many facets can the polytope $ \{x\in\RR^n\ :\ \|D^T x\|_1\leq 1\}$ ...

**5**

votes

**1**answer

144 views

### Measurement of “symmetry” of a convex body

I often hear that the regular simplex is "the least" symmetric convex body, and I've heard that there are some measures of symmetry of a body, that the simplex minimizes.
Could you please explain or ...

**9**

votes

**2**answers

251 views

### Book on the tetrahedron

Does anybody know of a book containing "all you want to know about the tetrahedron"? What you want to know should include basic geometry of the tetrahedron, study of orthocentric tetrahedra, the Monge ...

**22**

votes

**5**answers

703 views

### Is the diameter of a centrally symmetric convex body realized by a pair of antipodal points?

Let $S \subset \mathbb{R}^n$ be the boundary of a centrally symmetric convex body and provide $S$ with the geodesic metric given by its embedding in Euclidean space (i.e., the distance between two ...

**5**

votes

**2**answers

125 views

### Differentiability of polytope shadow areas

Let $P$ be an opaque convex polyhedron containing the origin in $\mathbb{R}^3$,
and let $S$ be an origin-centered sphere strictly containing $P$: $S \supset P$.
For a point $x$ on $S$, let $\sigma(x)$ ...

**0**

votes

**1**answer

110 views

### polynomial expression for counting number of integral points of a set

Let $v_i=a_ie_i\in\mathbb R^d$ and $w_i=b_ie_i\in\mathbb R^d$ for $i=1,\dots,d$ where $e_i$'s are unit vetcors and $a_i,b_i$ are positive integers. Let $$S=conv\{0,rv_i+sw_i:i=1,\dots,d\}.$$
Can we ...

**2**

votes

**1**answer

38 views

### Efficient algorithm for finding normals of a high dimensional convex hull with few facets

I am looking for the most efficient algorithm to use to, given a set of points in $d$ dimensional space, find the normals of the convex hull of these points, given that I know that the number of ...

**4**

votes

**2**answers

64 views

### Expressing a convex Polytope as a sublevel set of a function

Given an n-dimensional polytope $P$ in $\mathbb R^n$, Given as a convex hull of a finite set of points, $S$ I would like to construct an expliict formula for a function $f\colon \mathbb R^n \to ...

**2**

votes

**0**answers

129 views

### Find the intersection between two convex hulls, in this specific case

We work over $\mathbb{R}^K$. Let $V$ be the set of vectors whose coordinates take values $0$ or $1$, or equivalently the corners of the unit cube $[0,1]^K$.
Let $d:\{0, \ldots, K\} \to \mathbb{R}_+$ ...

**2**

votes

**0**answers

63 views

### Newton polyhedron and product of ideals

Let $I$ be an ideal generated by monomials $\underline{x}^{\underline{a}_1},\ldots ,\underline{x}^{\underline{a}_s}$ and
$J$ be the ideal generated by $\underline{x}^{\underline{b}_1},\ldots ...

**46**

votes

**5**answers

2k views

### Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?

Choose unit quaternions $q_0, q_1, q_2, q_3, q_4$ that form the vertices of a regular 4-simplex in the quaternions. Assume $q_0 = 1$. Let the other four generate a group via quaternion ...

**5**

votes

**1**answer

142 views

### Integer decomposition of dilated integral polytopes

For $n > 0$, let $P$ be an integral polytope, that is, the convex hull in $\mathbb{R}^n$ of points in $\mathbb{Z}^n$. Suppose that $\dim(P) = n$.
Question: Given $d > n + 2$ is it true that
$$ ...

**1**

vote

**1**answer

104 views

### Extreme points of convex hull of Minkowski sum [closed]

Let $\operatorname{conv}(a_1,\ldots,a_m)$ denote the convex hull of $\{a_1,\ldots,a_n\}$. Let $P = \operatorname{conv}(a_1,\ldots,a_p)$ and $Q = \operatorname{conv}(b_1,\ldots,b_q)$ be two convex sets ...

**3**

votes

**1**answer

195 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 ...

**8**

votes

**2**answers

467 views

### When does every point in a polytope lie along a chord between its edges?

Consider the 3-simplex, or tetrahedron, in 3-space. Regardless of the positions of the vertices, every point in the simplex lies on a chord between two non-adjacent edges of the simplex. Or, ...

**3**

votes

**1**answer

230 views

### Polar interpretation of convexity

Let $C$ be a convex polygon in the plane containing the origin, and let $r(\theta)$ for $\theta\in[0,2\pi)$ be a parametrization of its boundary. Is there a condition on $r$ that is equivalent to (or ...

**1**

vote

**0**answers

40 views

### Projection of a ray onto a random polytope

Suppose $P$ is a polytope formed by $p$ (general) random planes in $\mathbb{R}^n$. We assume $p \asymp n$ and $P$ has a diameter $O(\sqrt{n})$. For any $x \in \mathbb{R}^n$, denote by ...

**19**

votes

**1**answer

442 views

### Update to Shephard's “Twenty Problems on Convex Polyhedra”

Forty-three years ago, Geoffrey Shephard published an influential list of open problems
on convex polyhedra.
Progress has been made on several of his problems, and perhaps some have been completely ...

**5**

votes

**1**answer

128 views

### Approximation of convex body by polytopes

In a recent survey paper, Approximation of convex sets by polytopes, Bronstein claims that under Hausdorff distance $\rho_H$, for every convex body $U$,
$$\rho_H(U,\mathcal{P}_n)\leq c(U) ...

**2**

votes

**2**answers

168 views

### 4-polytope with vertices at the binary octahedral group

Does anybody know if there is a convex polytope in $R^4$ with vertices at the binary octahedral group (identitfying $H$ with $R^4$).
The binary tetrahedral group lies at the vertices of the ...

**1**

vote

**1**answer

85 views

### What is the functional form of the projections of a subdimensional polytope?

Let $\mathbf{S}$ be a $m\times n$ matrix, with $m < n$. We define a subdimensional polytope as the space of $n$-dimensional vectors $\mathbf{x}$ that satisfy the following equation:
...

**3**

votes

**1**answer

140 views

### Computionally efficient vertex enumeration for (convex) polytopes

Let $P \subseteq \mathbb{R}^d$ be an $\mathcal{H}$-polytope. The vertex enumeration problem asks for the set of vertices $V$ of $P$. Theoretically, the vertex enumeration problem for $P$ can be ...

**17**

votes

**5**answers

412 views

### How many unit simplices are needed to cover a unit $n$-cube?

The volume of an $n$-dimensional simplex of unit edge length is
$$V(n) = \frac{\sqrt{n+1}}{n! 2^{n/2}} \;,$$
so at least $\lceil 1/V(n) \rceil$ such simplices are needed to cover the unit $n$-cube.
...

**4**

votes

**1**answer

66 views

### Largest regular $k$-simplex inscribed in a $d$-cube, $k < d$

The largest (by edge length) regular simplex inscribed in a unit cube
is well known in $\mathbb{R}^2$ and $\mathbb{R}^3$:
Image sources:
left: NMSU,
right: Mathworld.
A recent ...

**7**

votes

**2**answers

185 views

### Decomposition of a cross-polytope into simplices

Consider the standard embedded $n$-cross polytope $P_n$ with vertices $\pm e_i \in \mathbb R^n$. Let us consider decompositions of this polytope into $2^{n-1}$ simplices, such that these simplices ...

**1**

vote

**1**answer

105 views

### Computation of extreme rays of rational polyhedral cones - Hemmecke's project and lift algorithm

I am working on an implementation of Raymond Hemmecke's algorithm for finding generating sets of cones: http://arxiv.org/abs/math/0203105
Unfortunately I am struggling to make the algorithm work on ...

**26**

votes

**14**answers

6k views

### Open problems in Euclidean geometry?

Which are some (research level) open problems in Euclidean geometry ?
(Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet)
I should clarify a ...

**16**

votes

**4**answers

784 views

### Is there a midsphere theorem for 4-polytopes?

The (remarkable) midsphere theorem says that each combinatorial
type of convex polyhedron may be realized by one all of whose edges are
tangent to a sphere
(and the realization is unique if the center ...

**1**

vote

**1**answer

68 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 = \{ ...

**38**

votes

**0**answers

1k views

### Intersecting Family of Triangulations

Let $\cal T_n$ be the family of all triangulations on an $n$-gon using $(n-3)$ non-intersecting diagonals. The number of triangulations in $\cal T_n$ is $C_{n-2}$ the $(n-2)$th Catalan number. Let ...

**13**

votes

**2**answers

264 views

### Can any simplex shadow-project to a regular simplex?

Every triangle $A$ can be oriented in $\mathbb{R}^3$
so that its orthogonal projection (shadow) onto the $xy$-plane is an
equilateral triangle $Q$:
...

**3**

votes

**0**answers

84 views

### Intersection numbers on blow ups of toric varieties

Suppose we have a smooth, complete toric varietiy $X_{\Sigma}$ of dimension $n$. Let $\sigma_k \in \Sigma(k)$ a smooth $k$-dimensional cone in $\Sigma$ and suppose we make the toric blow up at the ...

**3**

votes

**1**answer

126 views

### convex decompositions of the sphere

Consider a decomposition of the sphere $S^n$ into convex pieces (that is, every cell of the cell decomposition is convex, in particular, is contained in a hemisphere). Consider the $k$-skeleta of ...

**3**

votes

**0**answers

82 views

### Polytopes whose intersections have few vertices

This is related to one that I asked earlier:
The intersection of two $l_1$ balls
My question: are there any general classes of polytopes $P_1,P_2\subset\mathbb{R}^n$ such that $P_1$ has few ...

**2**

votes

**0**answers

87 views

### The intersection of two $l_1$ balls

Let $B_1$ and $B_2$ be two balls in $\mathbb{R}^n$ with respect to the $l_1$ norm that have different radii and different centers. Is there an upper bound for the number of vertices that $B_1\cap ...

**1**

vote

**0**answers

30 views

### Volume under the intersection of scaled simplices

This is rather specific but I need to compute the volume under the intersection of rescaled simplexes, that is, the volume of the space:
$\left\{x \in \mathbb{R}^n|\sum_i c_{ki} |x_i| \leq1\; k = 1 ...