Arrangements of homothets of a convex body II

Abstract

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 $2\cdot 3^d$ members. This improves a result of Polyanskii (arXiv:1610.04400). Using similar ideas, we also give a proof the following result of Polyanskii: Let $K_1,\dots,K_n$ 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 $n\leq O(3^d d)$.