# Arrangements of homothets of a convex body II

## TL;DR

This paper establishes upper bounds on the size of arrangements of homothets of convex bodies in Euclidean space, improving previous results and providing new bounds for specific configurations.

## Contribution

It improves bounds on the maximum size of Minkowski arrangements and related configurations of homothets of convex bodies, using novel geometric arguments.

## Key findings

- Any pairwise intersecting Minkowski arrangement has at most 2·3^d members.
- Sequences of homothets with centers on boundaries have at most O(3^d d) members.
- The results generalize and strengthen previous bounds by Polyanskii.

## 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 $n\leq O(3^d d)$.