Arrangements of Homothets of a Convex Body II




convex body, homothets, Minkowski arrangement, packing, covering


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).

Author Biographies

Marton Naszodi, Eotvos University, Budapest

Department of Geometry, Inst of Math. at Eotvos Univ., Budapest.

Konrad Swanepoel, London School of Economics and Political Science, UK

Department of Mathematics, London School of Economics and Political Science, UK


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