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

TL;DR

This paper explores properties of convex hull functions of convex bodies, proving key results about their determination, equivalences with classical problems, and applications to illumination bodies, with new insights into homothetic and translative volume properties.

Contribution

It establishes that convex hull functions do not determine the convex body, links the polar projection body problem with a conjecture on volume properties, and extends results to homothetic variants and illumination bodies.

Findings

Convex hull function does not uniquely determine the convex body.

Equivalence between the polar projection body problem and a volume conjecture.

Homothetic variant of the translative constant volume property for 3D convex polyhedra.

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 $K$ does not determine $K$. Furthermore, we prove the equivalence of the polar projection body problem raised by Petty, and a conjecture of G.Horv\'ath and L\'angi about translative constant volume property of convex bodies. We give a short proof of some theorems of Jer\'onimo-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.