Arrangements of homothets of a convex body

Answering a question of F"uredi and Loeb (1994), we show that the maximum number of pairwise intersecting homothets of a $d$-dimensional centrally symmetric convex body $K$, none of which contains the center of another in its interior, is at most $O(3^{d}d\log d)$. If $K$ is not necessarily centrally symmetric and the role of its center is played by its centroid, then the above bound can be replaced by $O(3^d\left(\genfrac{}{}{0px}{}{2d}{d}\right)d\log d)$. We establish analogous results for the case where the center is defined as an arbitrary point in the interior of $K$. We also show that in the latter case, one can always find families of at least $\mathrm{\Omega }((2\mathrm{/}\sqrt{3}{)}^d)$ translates of $K$ with the above property.