Metrics and Isometries for Convex Functions

TL;DR

This paper introduces new metrics for convex functions that align with epi-convergence and classifies their isometries, advancing the understanding of functional geometry.

Contribution

It develops a class of metrics for convex functions, proves their equivalence to epi-convergence, and classifies their isometries, extending geometric analysis tools.

Findings

Metrics are equivalent to epi-convergence.

Full classification of isometries for these metrics.

Introduction of new Hausdorff-like metrics for convex functions.

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. Furthermore, we give a full classification of all isometries with respect to some of the new metrics. Moreover, we introduce two 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.