Metrics and Isometries for Convex Functions

Abstract

Abstract We introduce a class of functional analogs of the symmetric difference metric on the space of coercive convex functions on ${\mathbb{R}}^n$ with full-dimensional domain. We show that convergence with respect to these metrics is equivalent to epi-convergence. For a large class of these natural metrics, we are able to provide a full classification of all isometries. In addition, we introduce new functional analogs of the Hausdorff metric on the spaces of coercive convex functions and super-coercive convex functions, respectively, and prove equivalence to epi-convergence.