Approximations of convex bodies by measure-generated sets

Abstract: We present a construction of convex bodies from Borel measures on ${\mathbb R}^n$. 
This construction allows us to study natural extensions of problems concerning the approximation 
of convex bodies by polytopes. In particular, we study a variation of the vertex index which, in 
a sense, measures how well a convex body can be inscribed into a polytope with small number of 
vertices. We discuss several estimates for these quantities, as well as an application to bounding 
certain average norms. Based on joint work with Han Huang.