Abstract
AbstractLetKbe a convex body which is (i) symmetric with respect to each of the coordinate hyperplanes and (ii) in isotropic position. We prove that most linear functionals acting onKexhibit super-Gaussian tail behavior. Using known facts about the mean-width of such bodies, we then deduce strong lower bounds for the volume of certain caps. We also prove a converse statement. Namely, if anarbitraryisotropic convex body (not necessarily satisfying (i)) exhibits similar cap-behavior, then one can bound its mean-width.
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