On the Hadwiger covering problem in low dimensions

A. Prymak; V. Shepelska

doi:10.1007/s00022-020-00554-3

paper

On the Hadwiger covering problem in low dimensions

arXiv:1811.08962 · doi:10.1007/s00022-020-00554-3

Abstract

Let $H_{n}$ be the minimal number of smaller homothetic copies of an $n$ -dimensional convex body required to cover the whole body. Equivalently, $H_{n}$ can be defined via illumination of the boundary of a convex body by external light sources. The best known upper bound in three-dimensional case is $H_{3} \leq 16$ and is due to Papadoperakis. We use Papadoperakis' approach to show that $H_{4} \leq 96$ , $H_{5} \leq 1091$ and $H_{6} \leq 15373$ which significantly improve the previously known upper bounds on $H_{n}$ in these dimensions.