WHAT IS A 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 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 Ω((2 / √ 3) d ) translates of K with the above property.