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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 byexternal 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 showthat $H_4\le 96$, $H_5\le 1091$ and $H_6\le 15373$ which significantly improvethe previously known upper bounds on $H_n$ in these dimensions.

On the Hadwiger covering problem in low dimensions