TY - GEN

T1 - Entropy and the hyperplane conjecture in convex geometry

AU - Bobkov, Sergey

AU - Madiman, Mokshay

PY - 2010/8/23

Y1 - 2010/8/23

N2 - 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.

AB - 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.

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U2 - 10.1109/ISIT.2010.5513619

DO - 10.1109/ISIT.2010.5513619

M3 - Conference contribution

AN - SCOPUS:77955679136

SN - 9781424469604

T3 - IEEE International Symposium on Information Theory - Proceedings

SP - 1438

EP - 1442

BT - 2010 IEEE International Symposium on Information Theory, ISIT 2010 - Proceedings

T2 - 2010 IEEE International Symposium on Information Theory, ISIT 2010

Y2 - 13 June 2010 through 18 June 2010

ER -