Convergence Properties of Convex Polyhedra

Abstract

Together with the Hausdorff metric, we consider two other metrics on the space of convex sets, namely, the metric induced by the Demyanov difference of convex sets and the Bartels–Pallaschke metric. We study the properties of convergence, continuity, and differentiability of convex polyhedral-valued mappings with respect to these three metrics. It is shown that the convergence of a sequence of convex sets to a convex polyhedron with respect to the Bartels–Pallaschke metric is equivalent to its convergence in the Demyanov metric