A Galerkin approach to optimization in the space of convex and compact subsets of R^d

The aim of this paper is to establish a theory of Galerkin approximations to the space of convex and compact subsets of $\R^d$ with favorable properties, both from a theoretical and from a computational perspective. These Galerkin spaces are first explored in depth and then used to solve optimization problems in the space of convex and compact subsets of $\R^d$ approximately.