ARRANGEMENTS OF HOMOTHETS OF A CONVEX BODY. Issue 2 (9th August 2017) - Naszódi, Márton; Pach, János; Swanepoel, Konrad
Abstract : Answering a question of Füredi and Loeb [On the best constant for the Besicovitch covering theorem. Proc. Amer. Math. Soc. 121 (4) (1994), 1063–1073], 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 2 d 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.