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

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.