  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.
