Arrangements of homothets of a convex body

Abstract

Answering a question of Füredi 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 $O(3^d(\genfrac{}{}{0ex}{}{2d}{d})d\log d)$. 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 $\Omega ((2/\sqrt{3}{)}^d)$ translates of $K$ with the above property.

13 pages, 2 figures. Accepted by Mathematika