Let K be 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 on K exhibit 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 an arbitrary isotropic convex body (not necessarily satisfying (i)) exhibits similar cap-behavior, then one can

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... und its mean-width. * This article is part of the author's Ph.D. thesis, written under the supervision of Professor Nicole Tomczak-Jaegermann at the University of Alberta. The author holds an Izaak Walton Killam Memorial Scholarship. involving the reverse inequality, namely super-Gaussian estimates of the form for t > 0 in some suitable range. Such estimates are garnering increased attention, as in [14] and [21] , and are the starting point for this paper. Our first main result concerns super-Gaussian directions for convex bodies that are isotropic and 1-unconditional. By isotropic, we mean that K has volume one, center of mass at the origin and each functional has the same variance, i.e.,