Covering the Crosspolytope with Crosspolytopes

Abstract

Let $\gamma^d_m(K)$ be the smallest positive number $\lambda$ such that the convex body $K$ can be covered by $m$ translates of $\lambda K$. Let $K^d$ be the $d$-dimensional crosspolytope. It will be proved that $\gamma^d_m(K^d)=1$ for $1\le m< 2d$, $d\ge4$; $\gamma^d_m(K^d)=\frac{d-1}{d}$ for $m=2d,2d+1,2d+2$, $d\ge4$; $\gamma^d_m(K^d)=\frac{d-1}{d}$ for $ m= 2d+3$, $d=4,5$; $\gamma^d_m(K^d)=\frac{2d-3}{2d-1}$ for $ m= 2d+4$, $d=4$ and $\gamma^d_m(K^d)\le\frac{2d-3}{2d-1}$ for $ m= 2d+4$, $d\ge5$. Moreover the Hadwiger's covering conjecture is verified for the $d$-dimensional crosspolytope.