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Bounds for Lattice Polytopes Containing a Fixed Number of Interior Points in a Sublattice

Published online by Cambridge University Press:  20 November 2018

Jeffrey C. Lagarias
Affiliation:
AT&T Bell Laboratories, Room 2C-373, Murray Hill, New Jersey, U.S.A. 07974
Günter M. Ziegler
Affiliation:
Universität Augsburg, Augsburg, Germany
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Abstract

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A lattice polytope is a polytope in whose vertices are all in . The volume of a lattice polytope P containing exactly k ≥ 1 points in d in its interior is bounded above by . Any lattice polytope in of volume V can after an integral unimodular transformation be contained in a lattice cube having side length at most n˙n ! V. Thus the number of equivalence classes under integer unimodular transformations of lattice poly topes of bounded volume is finite. If S is any simplex of maximum volume inside a closed bounded convex body K in having nonempty interior, then K ( n + 2)S — (n+ l)s where mS denotes a nomothetic copy of S with scale factor m, and s is the centroid of S.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1991

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