# 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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### 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 ...
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### Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?

Richard Stanley keeps a famous running compilation of different guises of the celebrated Catalan numbers. The number of vertices of the associahedron is one instantiation among the multitude, and the ...
2answers
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### Given the vertices of a convex polytope, how can we construct its half-space representation?

Let us say I have the vertices of a polytope $V = \{v_1,\dots,v_k\} \subset \mathbb R^n$. Is it possible to write $V$ as intersection of half-spaces using the information from the vertices, i.e., can ...
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### The $32$-deg polynomial for the tetrahedron inscribed in the icosahedron?

This MO answer discusses this table involving the maximal side lengths of the five Platonic solids $T,C,O,D,I$ inscribed in the other solids, This table is also found in Moritz Firsching's paper. I ...
3answers
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### 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) ...
2answers
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### 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 $X$. ...
1answer
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### Can the graph of a symmetric polytope have more symmetries than the polytope itself?

I consider convex polytopes $P\subseteq\Bbb R^d$ (convex hull of finitely many points) which are arc-transitive, i.e. where the automorphism group acts transitively on the 1-flags (incident vertex-...
3answers
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### Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?

Let $P$ be a convex polytope in $\mathbb{R}^d$ with $n$ vertices and $f$ facets. Let $\text{Proj}(P)$ denote the projection of $P$ into $\mathbb{R}^2$. Can $\text{Proj}(P)$ have more than $f$ facets? ...