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

The aim of this paper is to open up a new perspective on set and shape optimization by establishing 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. Galerkin spaces consisting of polytopes with fixed facet normals are first explored in depth and then used to solve optimization problems in the space of convex and compact subsets of R^d approximately.