On generalized Minkowski arrangements

Máté Kadlicskó,Zsolt Lángi

doi:10.26493/1855-3974.2550.d96

Abstract

The concept of a Minkowski arrangement was introduced by Fejes Tóth in 1965 as a family of centrally symmetric convex bodies with the property that no member of the family contains the center of any other member in its interior. This notion was generalized by Fejes Tóth in 1967, who called a family of centrally symmetric convex bodies a generalized Minkowski arrangement of order μ for some 0 < μ < 1 if no member K of the family overlaps the homothetic copy of any other member K′ with ratio μ and with the same center as K′. In this note we prove a sharp upper bound on the total area of the elements of a generalized Minkowski arrangement of order μ of finitely many circular disks in the Euclidean plane. This result is a common generalization of a similar result of Fejes Tóth for Minkowski arrangements of circular disks, and a result of Böröczky and Szabó about the maximum density of a generalized Minkowski arrangement of circular disks in the plane. In addition, we give a sharp upper bound on the density of a generalized Minkowski arrangement of homothetic copies of a centrally symmetric convex body.

Highlights

The notion of a Minkowski arrangement of convex bodies was introduced by L
Fejes Toth proved in [7] that the density of a Minkowski arrangement of circular disks in R2 with positive homogeneity is maximal for a Minkowski arrangement of congruent circular disks whose centers are the points of a hexagonal lattice and each disk contains the centers of six other members on its boundary
In [8] for any 0 < μ < 1 Fejes Toth defined a generalized Minkowski arrangements of order μ as a family F of centrally symmetric convex bodies with the property that for any two distinct members K, K′ of F, K does not overlap the μ-core of K′, defined as the homothetic copy of K′ of rat√io μ and concentric with K′

Summary

Introduction

The notion of a Minkowski arrangement of convex bodies was introduced by L. In [8] for any 0 < μ < 1 Fejes Toth defined a generalized Minkowski arrangements of order μ as a family F of centrally symmetric convex bodies with the property that for any two distinct members K, K′ of F , K does not overlap the μ-core of K′, defined as the homothetic copy of K′ of rat√io μ and concentric with K′ In this paper he made the conjecture that for any 0 < μ ≤ 3 − 1, the density of a generalized Minkowski arrangement of circular disks with positive homogeneity is maximal for a generalized Minkowski arrangement of congruent disks whose centers are the points of a hexagonal lattice and each disk touches the μ-core of six other members of the family.

Preliminaries

Remarks and open questions