On the convex hull and homothetic convex hull functions of a convex body
| Autor: | Ákos G. Horváth, Zsolt Lángi |
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| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: | |
| Popis: | 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. Comment: 12 pages, 1 figure |
| Databáze: | OpenAIRE |
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