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arXiv · 2012.08955

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

Ákos G. Horváth·Zsolt Lángi
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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.

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Ákos G. Horváth, Zsolt Lángi. 2020-12-16. On the convex hull and homothetic convex hull functions of a convex body. https://arxiv.org/abs/2012.08955

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