Generalization of Marion's Theorem: Volumes of Central Polytopes Obtained by Trisecting the Edges of Simplices

Abstract

The classical Marion's theorem states that the area of the central hexagon obtained by dividing each side of a triangle into three equal parts and connecting the interior division points is exactly $1/10$ of the area of the original triangle. In this paper, the construction is extended to an arbitrary n-dimensional simplex: each edge is divided into three equal parts, and through each n-1 vertices and the two interior division points of the opposite edge, two hyperplanes are drawn. It is shown that the resulting inner polytope has a surprisingly simple volume formula with the volume is $1/C(n,2n+1)$ times the volume of the original simplex. The proof uses a probabilistic interpretation with exponential random variables and reduces the geometric problem to a combinatorial binomial sum.