Asymptotics of generalized Hadwiger numbers

Details Asymptotics of generalized Hadwiger numbers Asymptotics of generalized Hadwiger numbers

2008-09-15

arxiv.org/abs/0809.2490v1

Mathematics

Metric Geometry

20p, 9 figures

Scientific paper

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.

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