NDLI: A proof of Lovász’s theorem on maximal lattice-free sets

Content Provider	Springer Nature Link
Author	Averkov, Gennadiy
Copyright Year	2012
Abstract	Let K be a maximal lattice-free set in $${\mathbb{R}^d}$$ , that is, K is convex and closed subset of $${\mathbb{R}^d}$$ , the interior of K does not contain points of $${\mathbb{Z}^d}$$ and K is inclusion-maximal with respect to the above properties. A result of Lovász asserts that if K is d-dimensional, then K is a polyhedron with at most 2$^{ d }$ facets, and the recession cone of K is a linear space spanned by vectors from $${\mathbb{Z}^d}$$ . A first complete proof of mentioned Lovász’s result has been published in a paper of Basu, Conforti, Cornuéjols and Zambelli (where the authors use Dirichlet’s approximation as a tool). The aim of this note is to give another proof of this result. Our proof relies on Minkowki’s first fundamental theorem from the geometry of numbers. We remark that the result of Lovász is relevant in integer and mixed-integer optimization.
Starting Page	105
Ending Page	109
Page Count	5
File Format	PDF
ISSN	01384821
Journal	Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry
Volume Number	54
Issue Number	1
e-ISSN	21910383
Language	English
Publisher	Springer-Verlag
Publisher Date	2012-02-11
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Cutting plane Lattice-free set Lovász’s theorem Mixed integer programming Integer programming Axiomatic and generalized convexity Lattices and convex bodies in $n$ dimensions Algebra Geometry Algebraic Geometry Convex and Discrete Geometry
Content Type	Text
Resource Type	Article
Subject	Algebra and Number Theory Geometry and Topology

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