Notes on lattice points of zonotopes and lattice-face polytopes

by Martin Henk, Matthias Henze, Eva Linke


Preprint series: 10-27

52C07 Lattices and convex bodies in $n$ dimensions, See Also {11H06, 11H31, 11P21}
52B20 Lattice polytopes (including relations with commutative algebra and algebraic geometry), See also {06A08, 13F20, 13Hxx}
52A40 Inequalities and extremum problems
11H06 Lattices and convex bodies, See also {11P21, 52C05, 52C07}


Abstract: Minkowski's second theorem on successive minima gives an upper bound on the volume of a convex body in terms of its successive minima. We study the problem to generalize Minkowski's bound by replacing the volume by the lattice point enumerator of a convex body. To this we are interested in bounds on the coefficients of Ehrhart polynomials of lattice polytopes via the successive minima. Our results for lattice zonotopes and lattice-face polytopes imply, in particular, that for 0-symmetric lattice-face polytopes and lattice parallelepipeds the volume can be replaced by the lattice point enumerator.

Keywords: Zonotope, lattice-face polytope, Ehrhart polynomial, successive minima

Notes: 15 pages

Upload: 2010-06-30

Update: 2011-01-18


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