[Banach] Abstract of a paper by Roman Vershynin

[Dale Alspach]

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 at math.ucdavis.edu

The paper may be downloaded from the archive by web browser from URL

 http://front.math.ucdavis.edu/math.FA/0403278

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 http://arXiv.org/abs/math.FA/0403278

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