Approximations of convex bodies by measure-generated sets

Problems pertaining to approximation and their applications have been extensively studied in the theory of convex bodies. In this talk we discuss several such problems, and focus on their extension to the realm of measures. In particular, we discuss variations of problems concerning the approximation of convex bodies by polytopes with a given number of vertices. This is done by introducing a natural construction of convex sets from Borel measures. We provide several estimates concerning these problems, and describe an application to bounding certain average norms.

Based on joint work with Han Huang