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.Horvath and Langi about translative constant volume property of convex bodies. We give a short proof of some theorems of Jeronimo-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.

Automated summary

The aim of this study is to investigate the properties of the convex hull and the homothetic convex hull functions of a convex body in Euclidean n-space. It is found that the convex hull function does not uniquely determine the convex body. Additionally, the study proves the equivalence between the polar projection problem and the translative constant volume property of convex bodies, and provides a brief proof of some theorems regarding the homothetic convex hull function. Furthermore, the study applies the results to describe the properties of the illumination bodies of convex bodies.