Arrangements of Homothets of a Convex Body II

Marton Naszodi, Konrad Swanepoel

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*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 Polyanskii:
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).


Keywords


convex body, homothets, Minkowski arrangement, packing, covering

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DOI: https://doi.org/10.11575/cdm.v13i2.62732

DOI (PDF): https://doi.org/10.11575/cdm.v13i2.62732.g46829

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Contributions to Discrete Mathematics. ISSN: 1715-0868