Classical Hahn-Banach Separation theorem plays a vital role in many branches of Analysis, Like functional Analysis, Convex Analysis, Variational Analyis, Theory of ODEs, optimal control and Optimization Theory etc...
Unfortunately the regularity conditions that satisfy the separation of two disjoint convex sets are highly restrictive in infinite dimensional spaces (like interiority and compactness conditions).
I am wondering do we have any other regularity-type conditions that guaranties separability of disjoint convex sets, or at least existence of supporting hyperplane at boundary points of convex sets?
I know there has been several effort to generalize interiority conditions like quasi-relative interiority .
Any help is much appreciated.
The only other separation theorem I have seen in active use is the Kreps-Yan theorem which states that a convex set $K$ containing the origin may be separated from a convex cone $C$ iff the closure of $K-C$ intersects $C$ only at the origin. This result is not universal to all Banach spaces, but it appears to be available in measure spaces, such as Orlicz spaces, see Proposition 3.5 in Gao, Niushan; Xanthos, Foivos, Option spanning beyond $L_p$ models, Math. Finan. Econ. 11, No. 3, 383–391 (2017).
Somewhat similar in spirit are the separation criteria found in these slides on p. 26 with a published reference Zălinescu, C., On the use of the quasi-relative interior in optimization, Optimization 64, No. 8, 1795-1823 (2015). ZBL1337.49060.
One can also note that Fenchel duality implies separation of certain convex sets and that necessary and sufficient condition for Fenchel duality to hold is lower semicontinuity of a certain perturbed functional, see Rockafellar's little book, Rockafellar, R.Tyrrell, Conjugate duality and optimization, CBMS-NSF Regional Conference Series in Applied Mathematics. 16. Philadelphia, Pa.: SIAM, Society for Industrial and Applied Mathematics. VI, 74 p. (1974). ZBL0296.90036, or in a more modern notation, Proposition 1.2 in Boţ, Radu Ioan, Conjugate duality in convex optimization, Lecture Notes in Economics and Mathematical Systems 637. Berlin: Springer (ISBN 978-3-642-04899-9/pbk; 978-3-642-04900-2/ebook). xii, 164 p. (2010). ZBL1190.90002.