Entropy and the hyperplane conjecture in convex geometry

Sergey Bobkov, Mokshay Madiman

Research output: Chapter in Book/Report/Conference proceedingConference contribution

4 Scopus citations

Abstract

The hyperplane conjecture is a major unsolved problem in high-dimensional convex geometry that has attracted much attention in the geometric and functional analysis literature. It asserts that there exists a universal constant c such that for any convex set K of unit volume in any dimension, there exists a hyperplane H passing through its centroid such that the volume of the section K ∩ H is bounded below by c. A new formulation of this conjecture is given in purely information-theoretic terms. Specifically, the hyperplane conjecture is shown to be equivalent to the assertion that all log-concave probability measures are at most a bounded distance away from Gaussianity, where distance is measured by relative entropy per coordinate. It is also shown that the entropy per coordinate in a log-concave random vector of any dimension with given density at the mode has a range of just 1. Applications, such as a novel reverse entropy power inequality, are mentioned.

Original languageEnglish (US)
Title of host publication2010 IEEE International Symposium on Information Theory, ISIT 2010 - Proceedings
Pages1438-1442
Number of pages5
DOIs
StatePublished - Aug 23 2010
Event2010 IEEE International Symposium on Information Theory, ISIT 2010 - Austin, TX, United States
Duration: Jun 13 2010Jun 18 2010

Publication series

NameIEEE International Symposium on Information Theory - Proceedings
ISSN (Print)2157-8103

Other

Other2010 IEEE International Symposium on Information Theory, ISIT 2010
CountryUnited States
CityAustin, TX
Period6/13/106/18/10

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