On the convex hull and homothetic convex hull functions of a convex body

Abstract

The aim of this note is to investigate the properties of the convex-hull and the homothetic convex-hull functions of a convex body K in Euclidean n-space, defined as the volume of the union of K and one of its translates, and the volume of K and a translate of a homothetic copy of K, respectively, as functions of the translation vector.In particular, we prove that the convex-hull function of the body K do not determine K. Furthermore, we prove the equivalence of the polar projection body problem raised by Petty, and a conjecture of G.Horváth and Lángi about translative constant volume property of convex bodies. We give a short proof of some theorems of Castro about the homothetic convex-hull function, and prove a homothetic variant of the translative constant volume property conjecture for 3-dimensional convex polyhedra.