Sections of the regular simplex - Volume formulas and estimates

TL;DR

This paper derives a general volume formula for intersections of regular simplices with subspaces, analyzes maximal and minimal volume sections, and provides bounds for k-dimensional sections, advancing geometric understanding of simplices.

Contribution

It introduces a new general volume formula for simplex sections and extends known results on maximal and minimal hyperplane sections using a novel proof technique.

Findings

Maximal volume hyperplane sections contain n-1 vertices.

For small distances to the centroid, hyperplanes with n-1 vertices maximize volume.

Provides bounds for k-dimensional sections.

Abstract

We state a general formula to compute the volume of the intersection of the regular $n$-simplex with some $k$-dimensional subspace. It is known that for central hyperplanes the one through the centroid containing $n-1$ vertices gives the maximal volume. We show that, for fixed small distances of a hyperplane to the centroid, the hyperplane containing $n-1$ vertices is still volume maximizing. The proof also yields a new and short argument for the result on central sections. With the same technique we give a partial result for the minimal central hyperplane section. Finally, we obtain a bound for $k$-dimensional sections.