On the volume of the convex hull of two convex bodies

Ákos G. Horváth, Z. Lángi

Published 2013-02-25, updated 2013-06-10Version 3

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.