The asymmetric sandwich theorem

TL;DR

This paper generalizes the Hahn-Banach theorem through the asymmetric sandwich theorem, deriving new duality results for affine functions on convex sets, applicable in various topological vector space contexts.

Contribution

It introduces the asymmetric sandwich theorem and extends Fenchel duality to affine functions on convex subsets, including non-locally convex spaces.

Findings

Derived bivariate, trivariate, quadrivariate duality results

Established duality under boundedness and Baire's theorem hypotheses

Extended duality theory to non-locally convex metrizable spaces

Abstract

We discuss the asymmetric sandwich theorem, a generalization of the Hahn-Banach theorem. As applications, we derive various results on the existence of linear functionals that include bivariate, trivariate and quadrivariate generalizations of the Fenchel duality theorem. Most of the results are about affine functions defined on convex subsets of vector spaces, rather than linear functions defined on vector spaces. We consider both results that use a simple boundedness hypothesis (as in Rockafellar's version of the Fenchel duality theorem) and also results that use Baire's theorem (as in the Robinson-Attouch-Brezis version of the Fenchel duality theorem). This paper also contains some new results about metrizable topological vector spaces that are not necessarily locally convex.