Seminar

Small ball probabilities (or anti-concentration inequalities) typically involve different tools  than those used for proving large deviation or concentration inequalities.   I will discuss a general method that leads to sharp small ball probabilities for quantities involving random vectors, their marginals, operator norms of random matrices and the volume of random convex sets. Various isoperimetric inequalities in convex geometry play a central role.  In particular,  stronger empirical versions in which stochastic dominance holds are well suited for small ball probabilities.  I will describe how techniques from convexity interface well with rearrangement inequalities such as the Rogers-Brascamp-Lieb-Luttinger inequality and Christ's version of the latter. (Based on joint work with G. Paouris).