Asymptotics of generalized Hadwiger numbers
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 ∂ 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.
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