On generalized Minkowski arrangements

TL;DR

This paper establishes sharp upper bounds on the total area and density of generalized Minkowski arrangements of circular disks and convex bodies in the plane, extending previous results and unifying concepts in geometric arrangements.

Contribution

It provides the first sharp upper bounds for the total area and density of generalized Minkowski arrangements of circular disks and convex bodies in the plane.

Findings

Sharp upper bound on total area of arrangements of circular disks.

Sharp upper bound on density of arrangements of homothetic convex bodies.

Unification and extension of previous Minkowski arrangement results.

Abstract

The concept of a Minkowski arrangement was introduced by Fejes T\'oth 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 T\'oth in 1967, who called a family of centrally symmetric convex bodies a generalized Minkowski arrangement of order $\mu$ for some $0 < \mu < 1$ if no member $K$ of the family overlaps the homothetic copy of any other member $K'$ with ratio $\mu$ 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 $\mu$ of finitely many circular disks in the Euclidean plane. This result is a common generalization of a similar result of Fejes T\'oth for Minkowski arrangements of circular disks, and a result of B\"or\"oczky and Szab\'o 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.