Applications of Convex Analysis within Mathematics

TL;DR

This paper explores the theoretical applications of convex analysis in optimization, monotone operator theory, and mathematical analysis, highlighting new proofs and deep connections within these areas.

Contribution

It introduces novel proofs of classical theorems and discusses advanced applications of convex analysis, especially infimal convolution, in various mathematical contexts.

Findings

Recovers Minty surjectivity theorem in Hilbert spaces

Provides a new proof of the sum theorem in reflexive spaces

Discusses autoconjugate representers for maximally monotone operators

Abstract

In this paper, we study convex analysis and its theoretical applications. We first apply important tools of convex analysis to Optimization and to Analysis. We then show various deep applications of convex analysis and especially infimal convolution in Monotone Operator Theory. Among other things, we recapture the Minty surjectivity theorem in Hilbert space, and present a new proof of the sum theorem in reflexive spaces. More technically, we also discuss autoconjugate representers for maximally monotone operators. Finally, we consider various other applications in mathematical analysis.