On an extremal problem connected with simplices

TL;DR

This paper explores the maximum volume of the convex hull formed by two intersecting congruent simplices in Euclidean space, providing new inequalities and exact results for regular simplices related by reflection.

Contribution

It introduces new inequalities and exact volume calculations for convex hulls of intersecting simplices, especially focusing on regular simplices and reflection symmetries.

Findings

Derived inequalities for the volume of convex hulls of intersecting simplices.

Determined the maximal volume for regular simplices related by reflection.

Established equalities connecting simplex configurations and their convex hulls.

Abstract

In this note we investigate the behavior of the volume that the convex hull of two congruent and intersecting simplices in Euclidean $n$-space can have. We prove some useful equalities and inequalities on this volume. For the regular simplex we determine the maximal possible volume for the case when the two simplices are related to each other via reflection at a hyperplane intersecting them.