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Let Z be an n-dimensional Gaussian vector and let f : R n → R be a convex function. We prove that:for all t > 1 where c > 0 is an absolute constant. As an application we derive variance-sensitive small ball probabilities for Gaussian processes.

We introduce and initiate the study of new parameters associated with any norm and any log-concave measure on R n , which provide sharp distributional inequalities. In the Gaussian context this investigation sheds light to the importance of the statistical measures of dispersion of the norm in connection with the local structure of the ambient space. As a byproduct of our study, we provide a short proof of Dvoretzky's theorem which not only supports the aforementioned significance but also complements the classical probabilistic formulation.

Abstract. We provide general inequalities that compare the surface area S(K) of a convex body K in R n to the minimal, average or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of K. We examine separately the dependence of the constants on the dimension in the case where K is in some of the classical positions or K is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.

The concentration of measure phenomenon in Gauss' space states that every L-Lipschitz map f on R n satisfies2L 2 , t > 0, where γn is the standard Gaussian measure on R n and M f is a median of f . In this work, we provide necessary and sufficient conditions for when this inequality can be reversed, up to universal constants, in the case when f is additionally assumed to be convex. In particular, we show that if the variance Var(f ) (with respect to γn) satisfies αL Var(f ) for some 0 < α 1, thenwhere c, C > 0 are constants depending only on α.