NDLI: Application of an idea of Voronoĭ to lattice packing, supplement

Content Provider	Springer Nature Link
Author	Gruber, Peter M.
Copyright Year	2015
Abstract	Voronoĭ’s criterion on lattice packings of balls in $$\mathbb E^d$$ of local maximum density was extended in an earlier article to lattice packings of convex bodies in $$\mathbb E^d$$ with local ultra maximum density, a stronger version of maximum density. No example of such a packing was given. However, it was conjectured that for generic convex bodies, the lattice packings of local maximum density are of local ultra maximum density provided that the dimension $$d$$ is sufficiently high. For generic convex bodies, the kissing number of lattice packings of global maximum density is between $$d(d+1)$$ and $$2d^2$$ . We conjecture that it equals $$2d^2$$ . In this note, first steps toward these conjectures are made. The existence of convex bodies with lattice packings of local ultra maximum density is shown. Contrasting this, families of convex bodies are specified which do not admit lattice packings of local ultra maximum density. If for a dense family of convex bodies every lattice packing of global maximum density has ultra maximum density, then this holds for a dense open family of convex bodies as well. A related result treats kissing numbers.
Ending Page	487
Page Count	15
Starting Page	473
File Format	PDF
ISSN	03733114
e-ISSN	16181891
Journal	Annali di Matematica Pura ed Applicata
Issue Number	2
Volume Number	195
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2015-01-29
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Quadratic forms (reduction theory, extreme forms, etc.) Kissing number Lattice packing and covering Lattices and convex bodies Mathematics Voronoĭ type result Extreme density Eutactic lattice Packing and covering Lattice packing of convex bodies Perfect lattice Extreme lattice Packing and covering in $n$ dimensions
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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