Asymptotics of generalized Hadwiger numbers

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.

💡 Research Summary

The paper investigates the asymptotic behavior of generalized Hadwiger numbers in the plane when the reference shape is a centrally symmetric oval (B) rather than a Euclidean disk. For a bounded planar domain (F) whose boundary (\partial F) is rectifiable, the authors define (N_B(F,\varepsilon)) as the maximal number of non‑overlapping homothetic copies of (\varepsilon B) that each intersect (F). The main theorem shows that as (\varepsilon\to0) the quantity satisfies
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