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Lattice points in lattice polytopes
Author(s) -
Pikhurko Oleg
Publication year - 2001
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300014339
Subject(s) - polytope , mathematics , simplex , combinatorics , lattice (music) , asymmetry , integer lattice , uniform k 21 polytope , dimension (graph theory) , lattice plane , upper and lower bounds , geometry , mathematical analysis , condensed matter physics , reciprocal lattice , convex set , regular polygon , physics , quantum mechanics , acoustics , diffraction , convex optimization , half integer
It is shown that, for any lattice polytope P ⊂ℝ d the set int ( P )∩lℤ d (provided that it is non‐empty) contains a point whose coefficient of asymmetry with respect to P is at most 8 d ⋅( 8 l + 7 )2 2 d + 1. If, moreover, P is a simplex, then this bound can be improved to 8 ⋅( 8 l + 7 )2 d + 1. As an application, new upper bounds on the volume of a lattice polytope are deduced, given its dimension and the number of sublattice points in its interior.