Mathematika

Table of Contents - July 2012 - Volume 58, Issue 02  

Research Article

ON THE EXISTENCE OF SUPERGAUSSIAN DIRECTIONS ON CONVEX BODIES

Grigoris Paourisa1

a1 Department of Mathematics, Texas A & M University, College Station, TX 77843, U.S.A. (email: grigoris@math.tamu.edu)

Abstract

We study the question of whether every centred convex body K of volume 1 in ℝ n has “supergaussian directions”, which means θ∈S n−1 such that

\[ \biggl |\biggl \{ x\in K: | \langle x, \theta \rangle |\gr t \int _{K} |\langle x, \theta \rangle | d x \biggr \}\biggr | \gr \mathrm {e}^{-c t^{2}}, \]

for all $ 1\ls t \ls \sqrt {n} $, where c>0 is an absolute constant. We verify that a “random” direction is indeed supergaussian for isotropic convex bodies that satisfy the hyperplane conjecture. On the other hand, we show that if, for all isotropic convex bodies, a random direction is supergaussian then the hyperplane conjecture follows.

(Received June 08 2011)

(Online publication November 24 2011)

MSC (2000)

  • 52A23 (primary)

Footnotes

The author is partially supported by NSF grant (DMS-0906150).