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 for all , 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.
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