# Tagged Questions

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### 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 ...
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### 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) ...
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### Does there always exist a self dual polytope that contains a given polytope contained in its dual?

Suppose a polytope $P$ is contained in its dual polytope $\tilde{P}$. Does there always exist a polytope $Q$ that contains $P$ and is self dual $Q=\tilde{Q}$? Is there any bound on the minimal number ...
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### Realization spaces for regular convex polytopes

Q1. Are there convex polytopes combinatorially equivalent to each of the regular polytopes that are realized with integer vertex coordinates? ...
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### Can all convex polytopes be realized with vertices on surface of convex body?

The following question was asked by me on Mathematics.SE. Unfortunately, no one answered it so I thought I might give it a try one level higher. Below the line you can find the slightly edited ...
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### Is it possible to sample uniformly on the surface of a high-dimensional polytope?

There are some pretty simple methods to do uniform sampling on the surface of high-dimensional spheres or cubes. Are there any methods that sample uniformly on the surface of a high-dimensional ...
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### Minimum solid angle and aspect ratio of an $n$-simplex

In computational geometry and other fields, it is of interest to have degeneracy measures for shapes of simplices, which quantitatively seperate the regular simplex from degenerate simplices. In two ...
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### Vertex-transitive polytopes in any dimension with any number of vertices?

Given positive integers $d$ and $v$ with $v \geq d+1$, does there always exist a (convex) vertex-transitive $d$-polytope with $v$ vertices? It seems that the answer should be "obviously" true, but I ...
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### Finding the convex combination of vertices which yields an inner point of a polytope

Given a convex polytope $P\in \mathbb{R}^n$, and a point $x\in P$, Caratheodory's theorem gives us that there exists a set of at most $n+1$ vertices of $P$, such that $x$ is a convex combination of ...
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### 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. ...
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### Convex n-polytope general position vectors to general position vectors of tetrahedron.

I asked this question in a comment to this question, but got no response. I thought that perhaps it needed more exposure, so I made it a question in itself. Define a set of general position vectors ...
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### Break Polyhedron into Tetrahedron

Given a polyhedron consists of a list of vertices (v), a list of edges (e), and a list of surfaces connecting those edges (...