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

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 does 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 Jerónimo-Castro about the homothetic convex hull function, and prove a homothetic variant of the translative constant volume property conjecture for 3-dimensional convex polyhedra. We also apply our results to describe the properties of the illumination bodies of convex bodies.