Let K ? Rn be a convex body of volume one. Let X1, ..., XN be independent random vectors distributed uniformly in K and let KN be their (symmetric) convex hull. A result of Groemer's states that the expected volume of KN is smallest when K is the Euclidean ball of volume one. A similar result, due to Bourgain, Meyer, Milman and Pajor, holds for the volume of random zonotopes ZN=?i=1N Xi. If T:RN?Rn is the (random) linear operator defined by Tei=Xi, for i=1, ..., N, then KN is the image of the unit ball in l1N, while ZN is the image of the unit ball in l8N. What happens when T is applied to other sets? I will discuss a unified approach to various inequalities involving the volume of random convex sets for which the Euclidean ball is the minimizer.