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Finite Lattice Packings and the Wulff-Shape

Part of: Discrete geometry

Published online by Cambridge University Press:  21 December 2009

Ulrich Betke  and
Károly Böröczky Jr.
Ulrich Betke
Affiliation:
Mathematisches Institut, Universität Siegen, D-57068 Siegen, Germany. E-mail: betke@mathematik.uni-siegen.d400.de
Károly Böröczky Jr.
Affiliation:
Math Inst. Hung. Acad. Sci., Budapest Pf. 127, 1364Hungary. E-mail: carlos@math-inst.hu
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Abstract

This paper treats finite lattice packings Cn + K of n copies of some centrally symmetric convex body K in Ed for large n. Assume that Cn is a subset of a lattice Λ, and ϱ is at least the covering radius; namely, Λ + ϱK covers the space. The parametric density δ(Cn, ϱ) is defined by δ(Cn, ϱ) = n · V(K)/V(convCn + ϱK). It is shown that, if δ(Cn, ϱ) is minimal for n large, then the shape of conv Cn is approximately given by Wulff's condition, well-known from crystallography. Thus maximizing parametric density is equivalent to optimizing a certain Gibbs–Curie energy. It is also proved that, in case of lattice packings of K (allowing any packing lattice), for large n the optimal shape with respect to the parametric density is approximately a Wulff-shape associated to some densest packing lattice of K.

MSC classification

Secondary: 52C17: Packing and covering in $n$ dimensions
Type
Research Article
Information
Mathematika , Volume 52 , Issue 1-2 , December 2005 , pp. 17 - 29
Copyright
Copyright © University College London 2005

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