Embeddings of infinitely connected planar domains into C^2

We prove that every circled domain in the Riemann sphere admits a proper holomorphic embedding to C^2. Our methods also apply to circled domains with punctures, provided that all but finitely many of the punctures belong to the closure of the set of complementary discs.Comment: Analysis and PDE, to appea

Finite convex geometries of circles

Let F be a finite set of circles in the plane. We point out that the usual convex closure restricted to F yields a convex geometry, that is, a combinatorial structure introduced by P. H Edelman in 1980 under the name "anti-exchange closure system". We prove that if the circles are collinear and they are arranged in a "concave way", then they determine a convex geometry of convex dimension at most 2, and each finite convex geometry of convex dimension at most 2 can be represented this way. The proof uses some recent results from Lattice Theory, and some of the auxiliary statements on lattices or convex geometries could be of separate interest. The paper is concluded with some open problems.Comment: 22 pages, 7 figure

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.Comment: 20p, 9 figure