This is an announcement for the paper "Volume of the polar of random sets and shadow systems" by Dario Cordero-Erausquin, Matthieu Fradelizi, Grigoris Paouris, and Peter Pivovarov. Abstract: 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. Archive classification: math.FA Submitted from: pivovarovp@missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1311.3690 or http://arXiv.org/abs/1311.3690