  We discuss isoperimetric inequalities for convex sets. These include the
classical isoperimetric inequality and that of Brunn-Minkowski,
Blaschke-Santalo, Busemann-Petty and their various extensions. We show that
many such inequalities admit stronger randomized forms in the following sense:
for natural families of associated random convex sets one has stochastic
dominance for various functionals such as volume, surface area, mean width and
others. By laws of large numbers, these randomized versions recover the
classical inequalities. We give an overview of when such stochastic dominance
arises and its applications in convex geometry and probability.
