Maximal monotonicity and cyclic involutivity of multi-conjugate convex functions

TL;DR

This paper explores the properties of multi-conjugate convex functions, focusing on cyclic involutivity and maximal monotonicity, revealing that these properties hold on the real line but generally fail in higher dimensions without additional regularity assumptions.

Contribution

It extends the understanding of involution and maximal monotonicity from dual pairs to multiple convex functions, highlighting differences between one-dimensional and multi-dimensional spaces.

Findings

Cyclical involutivity holds on the real line for multi-conjugate convex functions.

In higher dimensions, these properties generally do not hold without extra regularity.

Examples illustrate key differences between dual and multi-conjugate convex functions.

Abstract

A cornerstone in convex analysis is the crucial relationship between functions and their convex conjugate via the Fenchel-Young inequality. In this dual variable setting, the maximal monotonicity of the contact set $ \big\{(x,y) \ \big| \ f(x) + f^*(y) = \langle x,y \rangle \big\}$ is due to the involution $f^{**} = f$ holding for convex lower-semicontinuous functions defined on any Hilbert space. We investigate the validity of the cyclic version of involution and maximal monotonicity for multiple (more than two) convex functions. As a result, we show that when the underlying space is the real line, cyclical involutivity and maximal monotonicity induced by multi-conjugate convex functions continue to hold as for the dual variable case. On the other hand, when the underlying space is multidimensional, we show that the corresponding properties do not hold in general unless a further regularity assumption is imposed. We provide detailed examples that illustrate the significant differences between dual- and multi-conjugate convex functions, as well as between uni- and multi-dimensional underlying spaces.