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arXiv · 1312.4411

A point in a $nd$-Polytope is the barycenter of $n$ points in its $d$-faces

Michael Gene Dobbins
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Abstract

Using equivariant topology, we prove that it is always possible to find $n$ points in the $d$-dimensional faces of a $nd$-dimensional convex polytope $P$ so that their center of mass is a target point in $P$. Equivalently, the $n$-fold Minkowski sum of a $nd$-polytope's $d$-skeleton is that polytope scaled by $n$. This verifies a conjecture by Takeshi Tokuyama.

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Michael Gene Dobbins. 2014-06-05. A point in a $nd$-Polytope is the barycenter of $n$ points in its $d$-faces. https://doi.org/10.1007/s00222-014-0523-2

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