Integer cells in convex sets

Details Integer cells in convex sets Integer cells in convex sets

2004-03-16

arxiv.org/abs/math/0403278v2

Mathematics

Functional Analysis

Historical remarks on the notion of the combinatorial dimension are added. This is a published version in Advances in Mathemat

Scientific paper

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 Vapnik-Chervonenkis 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.

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