Annali di Matematica Pura ed Applicata (1923 -)

, Volume 193, Issue 4, pp 939–959 | Cite as

Application of an idea of Voronoĭ to lattice packing

  • Peter M. GruberEmail author


The geometric variant of a criterion of Voronoĭ says, a lattice packing of balls in \(\mathbb{E }^d\) has (locally) maximum density if and only if it is eutactic and perfect. This article deals with refinements of Voronoĭ’s result and extensions to lattice packings of smooth convex bodies. Versions of eutaxy and perfection are used to characterize lattices with semi-stationary, stationary, maximum and ultra-maximum lattice packing density, where ultra-maximality is a sharper version of maximality. Surprisingly, for balls, the lattice packings with maximum density have ultra-maximum density. To make the picture more complete, for \(d=2,3\), we specify the lattices that provide lattice packings of balls with maximum properties. These lattices are related to Bravais types. Finally, similar results of a duality type are given.


Voronoĭ type result Lattice packing of balls Lattice packing of convex bodies Maximum density Maximum lattice Perfect lattice Eutactic lattice 

Mathematics Subject Classification (2000)

05B40 11H06 11H31 11H55 52C07 52C17 



For valuable advice and hints, the author is obliged to the crystallographers Peter Engel and Erich Zobetz, and to Peter McMullen, Marjorie Senechal, and Tony Thompson.


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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institut für Diskrete Mathematik und GeometrieTechnische Universität WienViennaAustria

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