  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.
