ON THE EXISTENCE OF SUPERGAUSSIAN DIRECTIONS ON CONVEX BODIES
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We study the question of whether every centred convex body K of volume 1 in n has "supergaussian directions", which means S n-1 such that |[{x K: | x, | t int: K | x, | d x \biggr | -ct2 for all 1 t 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. 2012 Copyright University College London.