ARRANGEMENTS OF HOMOTHETS OF A CONVEX BODY II

A family of homothets of an o-symmetric convex body K in d-dimensional Euclidean space is called a Minkowski arrangement if no homothet contains the center of any other homothet in its interior. We show that any pairwise intersecting Minkowski arrangement of a d-dimensional convex body has at most 2 . 3(d) members. This improves a result of Polyanskii (Discrete Mathematics 340 (2017), 1950-1956). Using similar ideas, we also give a proof the following result of Polyan- skii: Let , K-1, ... ,K-n be a sequence of homothets of the o-symmetric convex body K, such that for any i < j, the center of K-j lies on the boundary of K-i. Then n = O(3(d)d)

Coloring translates and homothets of a convex body

We obtain improved upper bounds and new lower bounds on the chromatic number as a linear function of the clique number, for the intersection graphs (and their complements) of finite families of translates and homothets of a convex body in \RR^n.Comment: 11 pages, 2 figure

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

Arrangements of homothets of a convex body II

A family of homothets of an o-symmetric convex body K in d-dimensional Euclidean space is called a Minkowski arrangement if no homothet contains the center of any other homothet in its interior. We show that any pairwise intersecting Minkowski arrangement of a ddimensional convex body has at most 2 · 3d members. This improves a result of Polyanskii (Discrete Mathematics 340 (2017), 1950–1956). Using similar ideas, we also give a proof the following result of Polyanskii: Let K1, . . . , Kn be a sequence of homothets of the o-symmetric convex body K, such that for any i < j, the center of Kj lies on the boundary of Ki. Then n = O(3dd)