# Questions tagged [convex-polytopes]

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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### Toric decomposition of multipartitions

Fix $k \in \mathbb Z_{>0}$. By a $k$-multipartition $\lambda=(\lambda_1,\dots,\lambda_k)$ of $N$, I mean that each $\lambda_i$ is a partition of some $N_i$ and $\sum N_i = N$. Let's call $\lambda$ ...
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### Can a polytopal graph be "centrally symmetric" in more than one way?

Let $P,Q$ be two centrally symmetric convex polytopes, potentially of different dimensions and combinatorial type, but with the same edge-graph $G$. The central symmetry of $P$ induces an involutory ...
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### Bounding the number of facets of a polytope to approximate a given convex shape in higher dimensions

We are given a convex shape $S$ lying inside the hypercube $[0,1]^d$ in the $d$-dimensional Euclidean space. Let the volume $V(S)$ of $S$ be $\tfrac12$ (I guess nothing changes for any other fixed ...
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### Known configurations maximizing the volume of the convex hull of n points on the unit sphere

For $n\geq 4$, let $V_n$ be the maximum volume of the convex hull of $n$ points on the unit sphere (in $\mathbb{R}^3$, although information on higher dimensions is welcome as well). I'm sure the ...
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### Covering the unit sphere in $\mathbf{R}^n$ with $2n$ congruent disks

Let $v_i$ be $2n$ points in $\mathbf{R}^n$, with equal distance $|v_i|$ from the origin. Suppose that the convex hull of these points contains the unit ball. Is it known that $|v_i|\geq\sqrt{n}$? ...
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### Intersection of the simplex with a space vector

The sets are defined in $\mathbb{R}_+^n$ $(n\geq 1)$. The relative interior of a convex set $C$ will be denoted $\mathring C$. The codimension of a submanifold $M$ of a manifold $N$ will be denoted ...
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### Given a group action on a simplex, can I always find a fundamental region that is a simplex?

Let $\Delta\subset\Bbb R^n$ be a simplex with $n+1$ vertices. Let $G\subset\mathrm{GL}(\Bbb R^n)$ be a finite group of linear symmetries of $\Delta$, i.e. linear transformations that fix the simplex ...
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### Algorithm to find a facet of a polyhedron given the vertices?

I have a set of points $X=\{x_1,\ldots,x_N\}$ on the unit sphere in $\mathbb{R}^d$ ($N>d\ge 3$). What is an algorithm to find any facet of the polyhedron whose vertices are $X$? The set $X$ has ...
1 vote
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### Volume ratio of polytopes with few vertices

The volume ratio of a convex body $K\subset \mathbb{R}^{n}$ is $v_r(K) = \inf_{\mathcal{E}\subset K} \left(\frac{Vol(K)}{Vol(\mathcal{E})}\right)^{1/n}$ where the infimum run over ellipsoids included ...