Asymptotics of generalized Hadwiger numbers

TL;DR

This paper provides asymptotic estimates for the number of non-overlapping homothetic copies of a symmetric oval intersecting a domain, linking boundary length in Minkowski metric to geometric arrangements, and extends rectifiability results.

Contribution

It introduces a method for sliding convex beads in Minkowski space and extends rectifiability theorems to Minkowski geometry.

Findings

Asymptotic estimates relate boundary length to packing of homothetic copies.

Method for sliding convex beads in Minkowski plane.

Level sets are rectifiable in Minkowski space.

Abstract

We give asymptotic estimates for the number of non-overlapping homothetic copies of some centrally symmetric oval $B$ which have a common point with a 2-dimensional domain $F$ having rectifiable boundary, extending previous work of the L.Fejes-Toth, K.Borockzy Jr., D.G.Larman, S.Sezgin, C.Zong and the authors. The asymptotics compute the length of the boundary $\partial F$ in the Minkowski metric determined by $B$. The core of the proof consists of a method for sliding convex beads along curves with positive reach in the Minkowski plane. We also prove that level sets are rectifiable subsets, extending a theorem of Erd\"os, Oleksiv and Pesin for the Euclidean space to the Minkowski space.