Integer cells in convex sets
Abstract
Every convex body K in R^n has a coordinate projection PK that contains at least vol(0.1 K) cells of the integer lattice PZ^n, provided this volume is at least one. Our proof of this counterpart of Minkowski's theorem is based on an extension of the combinatorial density theorem of Sauer, Shelah and VapnikChervonenkis to Z^n. This leads to a new approach to sections of convex bodies. In particular, fundamental results of the asymptotic convex geometry such as the Volume Ratio Theorem and Milman's duality of the diameters admit natural versions for coordinate sections.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 March 2004
 arXiv:
 arXiv:math/0403278
 Bibcode:
 2004math......3278V
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Combinatorics;
 52C07;
 46B07;
 05D05
 EPrint:
 Historical remarks on the notion of the combinatorial dimension are added. This is a published version in Advances in Mathematics