Georgia Institute of Technology College of Sciences

The focus of my talk will be stochastic forms of isoperimetric inequalities for convex sets. I will review some fundamental inequalities including the classical isoperimetric inequality and those of Brunn-Minkowski and Blaschke-Santalo on the product of volumes of a convex body and its polar dual. I will show how one can view these as global inequalities that arise via random approximation procedures in which stochastic dominance holds at each stage. By laws of large numbers, these randomized versions recover the classical inequalities. I will discuss when such stochastic dominance arises and its applications in convex geometry and probability. The talk will be expository and based on several joint works with G. Paouris, D. Cordero-Erausquin, M. Fradelizi, S. Dann and G. Livshyts.