International audienceWe 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–Santaló inequality which, in turn, can be derived by the law of large numbers. The method involves shadow systems, their connection to Busemann type inequalities, and how they interact with functional rearrangement inequalities
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