On the volume of the convex hull of two convex bodies

TL;DR

This paper investigates the volume of the convex hull formed by two congruent convex bodies under various isometries in Euclidean space, proving inequalities and confirming a conjecture by Rogers and Shephard.

Contribution

It provides new inequalities for the volume of convex hulls of congruent bodies and proves a conjecture related to these geometric configurations.

Findings

Established volume inequalities for convex hulls of translated bodies

Proved the Rogers and Shephard conjecture in specific cases

Analyzed convex hulls under reflections and translations

Abstract

In this note we examine the volume of the convex hull of two congruent copies of a convex body in Euclidean $n$-space, under some subsets of the isometry group of the space. We prove inequalities for this volume if the two bodies are translates, or reflected copies of each other about a common point or a hyperplane containing it. In particular, we give a proof of a related conjecture of Rogers and Shephard.