Quadrivariate existence theorems and strong representability

TL;DR

This paper establishes conditions for computing conjugates of convex functions on product Frechet spaces, providing simplified proofs and stability results for strongly representable multifunctions, with applications to Banach spaces.

Contribution

It introduces new conditions for conjugate computation on product Frechet spaces and applies these to derive simplified proofs and stability results for representable multifunctions.

Findings

Derived new conjugate computation conditions for convex functions.

Provided simplified proofs for existing and new results in Banach spaces.

Established stability results for strongly representable multifunctions.

Abstract

In this paper, we give conditions under which we can compute the conjugate of a convex function on the product of two Frechet spaces defined in terms of another convex function on the product of two (possibly different) Frechet spaces. We use this result to give simple proofs of some (both old and new) results for Banach spaces, and deduce some (both old and new) stability results for strongly representable multifunctions. We take as our starting point a result on closed convex cones in the product of two Frechet spaces.