Galerkin approximations to the space of convex bodies by polytopes in nondegenerate V-representation

Abstract: We introduce finite-dimensional approximations to the space of convex bodies based on polytopes in vertex representation. For a prescribed set of directions, the admissible point configurations form a polyhedral convex subcone of the Euclidean vector space, described by a finite system of linear inequalities. The interior of this cone contains only nondegenerate representations, in which all parameter points are distinct vertices of the represented polytope. We study the geometry of the parameter cone, redundancies in its defining inequalities, and natural projections of convex bodies onto the resulting polytope spaces. We derive quantitative approximation estimates in terms of how densely the prescribed directions cover the unit sphere and construct nested Galerkin sequences whose approximation error converges locally uniformly to zero. Finally, we use these approximation properties to establish the convergence of finite-dimensional approximations of constrained global optimization problems in the space of convex bodies.