  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.
