Nonlinear Fenchel Conjugates

TL;DR

This paper extends Fenchel conjugation to functions on arbitrary sets using nonlinear test functions, exploring properties like biconjugation and applications on manifolds and Lie groups.

Contribution

It introduces nonlinear Fenchel conjugation, generalizing classical concepts to broader settings without linear structure, and derives new results in these contexts.

Findings

Generalizes Fenchel conjugation to arbitrary sets

Establishes properties including Fenchel-Moreau biconjugation

Connects to convexity on Lie groups

Abstract

The classical concept of Fenchel conjugation is tailored to extended real-valued functions defined on linear spaces. In this paper we generalize this concept to functions defined on arbitrary sets that do not necessarily bear any structure at all. This generalization is obtained by replacing linear test functions by general nonlinear ones. Thus, we refer to it as nonlinear Fenchel conjugation. We investigate elementary properties including the Fenchel-Moreau biconjugation theorem. Whenever the domain exhibits additional structure, the restriction to a suitable subset of test functions allows further results to be derived. For example, on smooth manifolds, the restriction to smooth test functions allows us to state the Fenchel-Young theorem for the viscosity Fr\'echet subdifferential. On Lie groups, the restriction to real-valued group homomorphisms relates nonlinear Fenchel conjugation to infimal convolution and yields a notion of convexity.