  In 1957, Hadwiger made the famous conjecture that any convex body of
$n$-dimensional Euclidean space $\mathbb{E}^n$ can be covered by $2^n$ smaller
positive homothetic copies. Up to now, this conjecture is still open for all
$n\geq 3$. Denote by $\gamma_{m}(K)$ the smallest positive number $\lambda$
such that $K$ can be covered by $m$ translations of $\lambda K$. The values of
$\gamma_m(K)$ for some particular $m$ and $K$ have been studied. In this
article, we will focus on the situation where $K$ is the unit crosspolytope of
the three-dimensional.
