Variations on a theorem by Edwards

TL;DR

This paper explores two extensions of Edwards' duality theorem, broadening its applicability to cones with zero sets and introducing new functional and measure-based dualities, with applications in complex analysis.

Contribution

It presents two novel variations of Edwards' duality theorem, extending its scope and introducing new dualities involving continuous functionals and order relations.

Findings

Extended duality theorem to cones without constant functions

Replaced suprema and infima with new functionals and measures

Applied results to discontinuity propagation and minimal elements characterization

Abstract

We discuss two variations of Edwards' duality theorem. More precisely, we prove one version of the theorem for cones not necessarily containing all constant functions. In particular, we allow the functions in the cone to have a non-empty common zero set. In the second variation, we replace suprema of point evaluations and infima over Jensen measures by suprema of other continuous functionals and infima over a set measures defined through a natural order relation induced by the cone. As applications, we give some results on propagation of discontinuities for Perron--Bremermann envelopes in hyperconvex domains as well as a characterization of minimal elements in the order relation mentioned above.