Dual Pairs and maximality

Linear separation theorems, besides being important results in Convex Ananysis, play a central role in the proofs of several central theorems in other fields. Morover, several results as such can be proven to be equivalent in themselves to a suitable separation theorem. In this paper we start analysing such a phenomenon, showing that the "Dual Cone" theorem between dual pairs of linear spaces implies several separation results, and that it can be exploited for several purposes, such as chatacterising both maxima of convex sets and aolutions of convex optimisation problems.