**1**

vote

**2**answers

323 views

### Does an $n$ dimensional subspace intersect the $n$-facets of the unit cube?

Since my intuition for high dimensional geometry is not always right:
Consider the unit cube in $\mathbb{R}^m$ and for $n\leq m$ denote by $F^n$ the union of the $n$-facets. For what numbers of $m$ ...

**1**

vote

**1**answer

253 views

### Hypercube vertex sampling with largest convex cone

Maybe this question was already asked and forgive me if I can't formulate it well.
Lets assume we have a n-dimensional hypercube. If we want the smallest set of vertices such that the cube is inside ...

**1**

vote

**0**answers

146 views

### When is the conical hull of a finite set of vectors a subset of the space? (and tilings)

Consider a hypercube in n-dimensions, and take some projection down to an m-dimensional subspace. Now take all vertices and m-1 dimensional facets visible from some direction outside the projection. ...

**5**

votes

**2**answers

374 views

### Measure of intersection of polyhedral cone with unit sphere

Let $C$ be a pointed polyhedral cone in $\mathbb{R}^n$ and let $S^{n-1}$ denote the unit sphere in $\mathbb{R}^n$. Given a description of the supporting hyperplanes of $C$ is there an algorithm for ...

**3**

votes

**0**answers

144 views

### An isoperimetric inequality for “order” polytopes

I am looking for an isoperimetric inequality for order-like polytopes.
An order polytope $K\in \mathbb{R}^n$ is defined by a set of linear inequaities:
$$ \forall i \; 0\leq x_i \leq 1 $$
and
$ ...

**7**

votes

**2**answers

430 views

### When does every point in the volume of 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 volume defined by the triangular faces of the polytope's skeleton graph can lie ...

**8**

votes

**1**answer

1k views

### About the surface area vs. volume of polytopes

Given a convex body $K\in\mathbb{R}^n$, represented by a set of linear inequalities (intersection of halfspaces), I am interested in understanding how much of its volume can be close to its perimeter ...

**11**

votes

**2**answers

2k views

### Weighted area of a Voronoi cell

Let $X = \{ x_1,\dots,x_n\} $ denote a set of $n$ points in the unit square $S = [0,1]\times[0,1]$, and let $w = \{w_1,\dots,w_n\}$ denote a set of weights corresponding to the $n$ points in $X$. ...

**17**

votes

**0**answers

2k views

### Why do polytopes pop up in Lagrange inversion?

I'd be interested in hearing people's viewpoints on this. Looking for an intuitive perspective. See Wikipedia for descriptions of polytopes and the Lagrange inversion theorem/formula (LIF) for ...

**5**

votes

**1**answer

335 views

### Isometric embedding a convex cap to render its boundary planar

I would like to know if there is a polyhedral analog to this beautiful
theorem of Hong:
Theorem 11.0.1.
Any smooth positive disk $(\bar{D},g)$ with a positive geodesic
curvature along ...

**0**

votes

**0**answers

349 views

### Upper and Lower Bounds of the Total Surface Area of Convex Polytopes that Partition a Hypercube

Let $C$ be a hypercube in $\mathbb{R}^D$ with edge length of $L$. Let $\mathcal{P}_1,\ldots,\mathcal{P}_K$ be $K$ convex polytopes that partition $C$. Let $S_k$ be the surface area of the polytope ...

**5**

votes

**3**answers

2k views

### Random Sampling a linearly constrained region in n-dimensions…

Hi,
So here is my problem:
Given a nonlinear, discontinous, cost function $f(x_1,x_2,..,x_N)$ along with linear constraints $x_n \ge 0, \forall n$
$x_n \le c_n$
and $\sum_{n=1}^N x_n = 1$ find an ...

**2**

votes

**0**answers

215 views

### Sampling from a partition of a hypercube by convex polytopes.

I have a binary space partitioning (BSP) tree which recursively partitions a hypercube using linear hyperplanes. That is, a hyperplane splits the hybercube in half, creating two convex polytopes. Each ...

**1**

vote

**2**answers

441 views

### Extreme points of transportation polytope

I'm interested in $n \times m$ joint probability tables with prescribed row and column marginals. Such tables form a convex set known as the transportation polytope. What are the extreme points of ...

**2**

votes

**2**answers

500 views

### Finding smallest ellipsoid that circumscribes over intersection of two ellipsoids that do not have common center

Does this already exist in literature? The closest Ive been able to find is circumscribe intersection of two ellipsoids with a common center by W. Kahan ...

**2**

votes

**1**answer

290 views

### Angle btween Coordinate Vector and Normal Vector of Facet in a Convex Polytope, Asking for a Counterexample

Definitions
Let $\mathcal{C}$ be a convex polytope in $\mathbb{R}^{D}$ with $K$-facets
$F_{1},\ldots,F_{K}$. I denote the normal vector of the $k^\mathrm{th}$ facet as
...

**7**

votes

**3**answers

635 views

### Not quite regular polyhedra

Take a naive interpretation of regular polyhedra:
All vertices (including epsilon ball) congruent
All edges congruent
All faces congruent
We can now find interesting families by removing one ...

**9**

votes

**3**answers

738 views

### Polytopes with few vertices.

Suppose I have a convex polytope in $\mathbb{R}^d$ which I know has few vertices (in the case which prompted this question, I seem to have a polytope in $\mathbb{R}^9$ which has sixteen vertices). Is ...

**5**

votes

**1**answer

313 views

### Recovering a polyhedron from its tumble-density profile

Imagine a white convex polyhedron $P$ tumbling randomly about its fixed center of gravity (c.g.)
$c$ against a blue background.
A long-exposure photo would show pure white in a neighborhood of $c$
...

**12**

votes

**1**answer

637 views

### Optimal 8-vertex isoperimetric polyhedron?

I know from Marcel Berger's
Geometry Revealed:
A Jacob's Ladder to Modern Higher Geometry
(p.531)
that it is not yet established which polyhedron in $\mathbb{R}^3$ on 8 vertices achieves the optimal ...

**13**

votes

**2**answers

703 views

### Sampling from the Birkhoff polytope

The set of $n\times n$ real, nonnegative matrices whose rows and columns sum to one forms the well-known Birkhoff polytope
Recently someone asked me if I knew
How to sample (in polynomial time) ...

**8**

votes

**2**answers

687 views

### Regular polygon shadows of convex polyhedra

Fix a finite subset $S$ of the natural numbers $\mathbb{N}$, each element $\ge 3$.
Is there a convex polyhedron $P$ that has among its shadows
regular $n$-gons for each $n \in S$? Does such a ...

**1**

vote

**1**answer

208 views

### Representing integer points inside a polytope using a unit hypercube

Let $D\subset\mathbb{Z}\times\mathbb{Z}$ such that $D$ is formed by the points bounded by a convex polytope. Let $f:D\to\mathbb{R}$ defined by $f(x,y)=ax+by+c$, where $a,b,c\in\mathbb{R}$ and ...

**14**

votes

**1**answer

576 views

### Totally rational polytopes

Define a convex polytope in $\mathbb{R}^d$ as
totally rational (my terminology)
if its vertex coordinates are rational, its edge lengths
are rational, its two-dimensional face areas are rational, ...

**3**

votes

**2**answers

203 views

### time-shifted ODEs/volume of polytopes

Hello,
I'm looking for help with the following ODE:
f'(t) = x f(1 - at)
for 0 < a < 1, x in any interval (though 0 < x < 1 would be best), and f(0) = 1. There should be a solution for ...

**3**

votes

**1**answer

348 views

### Number of simplicial polytopes with a given f-vector

Plenty of very nice literature is available on the characterization of f-vectors of simplicial complexes of diverse sorts (results by Billera, Bjoerner, Kalai, Stanley, among others). I mention, as an ...

**5**

votes

**0**answers

182 views

### A polytope associated with the Hadamard Transform

In an investigation of whether or not a subset $V$ of "Hamming Space" $M_n = \mathbb{F}_2^n$ is a tile (i.e. whether $M_n$ can be written as a disjoint partition of translates of $V$) in ...

**6**

votes

**0**answers

262 views

### A question about a blue fan and a red fan and their common refinement

Is the following conjecture true?
Conjecture: Let $M_1$ be a red map and let $M_2$ be a blue map drawn in general position on $S^n$, and let $M$ be their common refinement. There is a vertex $w$ of ...

**5**

votes

**0**answers

393 views

### Reference for this polyhedral lemma

Recall the definition of a fan: Let $U$ be a finite dimensional real vector space. Then a fan is a collection $\mathcal{F}$ of cones in $U$ such that
(1) If $\sigma \in \mathcal{F}$ and $\tau$ is a ...

**1**

vote

**0**answers

156 views

### Marginals and Convex Sets

I am looking for a weak convergence of marginals result, in the following type of situation. References to 'related' situations are also very much appreciated.
I have a collection of affine ...

**4**

votes

**4**answers

537 views

### Parametrizing the realization space of a polyhedron by its edges

I alluded to this here, but at that point I hadn't really done enough work to know what I wanted to ask.
Call a polyhedron "trihedral" if three faces meet at each vertex. Each of the F faces can be ...

**5**

votes

**2**answers

419 views

### A convex polyhedral analog of the pentagram map

I am wondering if there is a three-dimensional analog of
the pentagram map, which maps a convex polygon to another
convex polygon. Here's the Wikipedia image:
...

**4**

votes

**1**answer

1k views

### intersection of convex and non-convex polyhedra

Hi everyone,
I am trying to find the best appropriate way to intersect polyhedra which may be non-convex.
The number of vertices that build the polyhedron is hence always small (up to 20 or so).
...

**13**

votes

**3**answers

667 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

326 views

### When is a complete fan a normal fan?

Is there a characterization for when a complete fan in $\mathbf{R}^n$ is the normal fan of a polytope? Thanks!

**8**

votes

**1**answer

746 views

### Finding the vertices of a polyhedral complex coming from a GIT wall and chamber decomposition

I am interested in a polyhedral/combinatorics problem that arises in algebraic geometry in the context of geometric invariant theory (GIT).
Algebro-geometric background: Consider the natural ...

**3**

votes

**0**answers

188 views

### Bell polytopes with nontrivial symmetries

Take $N$ parties, each of which receives an input $s_i \in {1, \dots, m_i}$ and produces an output $r_i \in {1, \dots, r_i}$, possibly in a nondeterministic manner. We are interested in joint ...

**4**

votes

**1**answer

397 views

### Embed the intersection of an n-dimensional unit $L_1$ sphere and a hyperplane into an (n-1)-dimensional unit $L_1$ sphere.

In $\mathbb{R}^n$, given an unit $L_1$ sphere $\mathcal{B}_n: |x_1|+|x_2|+\ldots+|x_n|\leq 1$ and a hyperplane $\mathcal{P}: a_1x_1+a_2x_2+\ldots+a_nx_n=0$. Does there always exist a rotation such ...

**3**

votes

**0**answers

216 views

### Maximum of a function on $d-$dimensional convex compact sets

Let $\mathcal C_d$ denote the set of all $d-$dimensional convex compact subsets with
barycenter at the origin of the $d$-dimensional Euclidean space $\mathbb E^d$. Given an
element $C\in\mathcal C_d$ ...

**12**

votes

**4**answers

639 views

### Name of a polytope

What is the name of the polytope $\Sigma\cap (-\Sigma)$ for $\Sigma$ a $d-$simplex with barycenter at the origin?
In dimension $2$, one gets a hexagon, in dimension $3$ an octahedron (given by the ...

**11**

votes

**0**answers

315 views

### Lower Bound on the Volume of Certain Polytopes

Given a partition $\rho\in\mathcal{P}(n)$ with $k$ blocks
$$
\rho=\{B_1,B_2,\ldots,B_{k}\}
$$
we can define the set of equations
$$
E_{i}:\sum_{j \in B_{i}}{x_{j-1}}=\sum_{j \in ...

**36**

votes

**4**answers

2k views

### Polynomial roots and convexity

A couple of years ago, I came up with the following question, to which I have no
answer to this day. I have asked a few people about this, most of my teachers and some
friends, but noone had ever ...

**2**

votes

**1**answer

165 views

### Subspace of $\mathbb{R}^n$ spanned by the image of convex $(n-1)$-polyhedra under the face-counting map

Fix $n \in \mathbb{N}$. A convex polyhedron $C$ in $\mathbb{R}^n$ is the convex hull of finitely many points with nonempty interior. For $H$ a supporting hyperplane, ie $C$ is contained in one of the ...

**16**

votes

**1**answer

659 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 $v$ ...

**0**

votes

**1**answer

168 views

### ask for the time complexity of a convex quadratically constrained quadratic program (QCQP) problem

Could someone tell me the time complexity of a convex quadratically constrained quadratic program (QCQP) problem? And any references?
Thank you very much.

**8**

votes

**3**answers

1k views

### Listing lattice points in a simplex

Let $n \in \mathbf{Z}_{\geq 1}$. Is there an algorithm which, given a simplex $\Delta \subset \mathbf{R}^n$ specified as the convex hull of $v_0,\dots,v_n \in \mathbf{Z}^n$, computes the set $\Delta ...

**6**

votes

**0**answers

483 views

### Counting Lattice Points in Real Polytopes

Suppose one did have an exact formula for the number of $\mathbb{Z}^n$-lattice points intersecting an arbitrary dilate of a (not necessarily rational) finite, closed and convex $n$-polytope. As a ...

**2**

votes

**1**answer

534 views

### An intersection of a convex polytope and a simplex

How to determine whether an intersection of a convex polytope and a simplex in $R^{n}$ is not empty?
The polytope is given in a halfspace representation.
I'm aware that there are some algorithms ...

**7**

votes

**1**answer

364 views

### The volume of the “unit ball” in $\mathbb{R}^{m\times n}$ with respect to the cut norm

This question is inspired by the question “ε-nets with respect to the cut norm” by the user Aaron, which had been reposted to cstheory.stackexchange.com.
The cut norm ||A||C of a matrix A=(aij)∈ℝm×n ...

**9**

votes

**2**answers

680 views

### What is determined by the combinatorics of the shadows of a convex polyhedron?

Define the shadow of a convex polyhedron $P$ in direction $u$
to be the orthogonal projection of $P$ onto a plane whose normal is $u$.
The shadow is a convex $k$-gon.
I am wondering to what degree $P$ ...