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(3ddlogd). 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(3d(2dd)dlogd). 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 Ω((2/3–√)d) translates of K with the above property.