Approximations of convex bodies by measure-generated sets

Given a Borel measure mu on Rn, we define a convex set by M(mu) = 0= f = 1, Rn f d mu= 1 Rn y f (y) d mu(y) , where the union is taken over mu-measurable functions f : Rn. [ 0, 1] such that Rn f d mu = 1 and Rn y f (y) d mu(y) exists. We study the properties of these measure-generated sets, and use them to investigate natural variations of problems of approximation of general convex bodies by polytopes with as few vertices as possible. In particular, we study an extension of the vertex index which was introduced by Bezdek and Litvak. As an application, we prove that for any non-degenerate probability measure mu, one has the lower bound Rn x Z1(mu) d mu(x) = c v n, where Z1 (mu) is the L1-centroid body of mu.