A Unified View of Polarity for Functions

TL;DR

This paper introduces a unified framework for understanding the polarity of functions, connecting various definitions and properties, and exploring new theoretical results including lattice isomorphisms and polar subdifferentials.

Contribution

It provides a comprehensive unification of polarity concepts for functions, generalizes existing properties, and establishes new theoretical connections and results.

Findings

Bipolar sets and bipolar functions form isomorphic lattices.

Three notions of polar subdifferential are introduced and connected to vector alignment.

The framework generalizes several known properties of function polarity.

Abstract

We propose a unified view of the polarity of functions, that encompasses all specific definitions, generalizes several well-known properties and provides new results. We show that bipolar sets and bipolar functions are isomorphic lattices. Also, we explore three possible notions of polar subdifferential associated with a nonnegative function, and we make the connection with the notion of alignement of vectors.