Gaussian mixtures with applications to Khinchine inequalities, entropy inequalities, and convex geometry.

We say that a symmetric random variable X is a Gaussian mixture if X has the same distribution as YG, where G is a standard Gaussian random variable, and Y is a positive random variable independent of G. We use this simple notion to study certain Schur convexity properties of p-th moments for weighted sums of independent random variables. This gives, in particular, optimal Kchinchine inequalities for Gaussian mixtures. The Shannon entropy of sums of independent random variables is also studied. In the second part of the talk we investigate, using Gaussian mixtures, several topics coming from convex geometry, such as the so-called B-inequality, correlation conjecture, and extremal sections and projections of B_p^n balls. We mention several open problems. Based on a joint work with Alexandros Eskenazis and Tomasz Tkocz.