  We obtain optimal inequalities for the volume of the polar of random sets,
generated for instance by the convex hull of independent random vectors in
Euclidean space. Extremizers are given by random vectors uniformly distributed
in Euclidean balls. This provides a random extension of the Blaschke-Santalo
inequality which, in turn, can be derived by the law of large numbers. The
method involves generalized shadow systems, their connection to Busemann type
inequalities, and how they interact with functional rearrangement inequalities.
