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arXiv · 1811.08962

On the Hadwiger covering problem in low dimensions

A. Prymak·V. Shepelska
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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 $H_3\le 16$ and is due to Papadoperakis. We use Papadoperakis' approach to show that $H_4\le 96$, $H_5\le 1091$ and $H_6\le 15373$ which significantly improve the previously known upper bounds on $H_n$ in these dimensions.

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A. Prymak, V. Shepelska. 2018-11-21. On the Hadwiger covering problem in low dimensions. https://doi.org/10.1007/s00022-020-00554-3

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