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).