This is an announcement for the paper "Integer cells in convex sets" by Roman Vershynin. Abstract: Every convex body K in R^n admits 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. Archive classification: Functional Analysis; Combinatorics Mathematics Subject Classification: 52C07, 46B07, 05D05 Remarks: 26 pages The source file(s), vr.tex: 57558 bytes, is(are) stored in gzipped form as 0403278.gz with size 18kb. The corresponding postcript file has gzipped size 89kb. Submitted from: vershynin@math.ucdavis.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0403278 or http://arXiv.org/abs/math.FA/0403278 or by email in unzipped form by transmitting an empty message with subject line uget 0403278 or in gzipped form by using subject line get 0403278 to: math@arXiv.org.