The weakest condition guarantees some Separation-type of convex sets in Banach spaces 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. 
 A: The following tiny mini-theory of convexity, which I've introduced in 1961, may serve as a start-point. It is abstract. However you may add an algebraic structure, topology, and whatever you wish, together with proper assumptions (e.g. local convexity in linear spaces) to get results in the respective environment.
Consider a pair $\ (X\ \mathcal C)\ $ consisting of a set $X$ and a family $\mathcal C$ of subsets of $X$. Call these subsets convex.
Definition 1   A set $\ H\subseteq X\ $ is called a half-space
$\ \ \Leftarrow:\Rightarrow
\ \ H\in\mathcal C\,\ \mbox{and}\,\ X\setminus H\in\mathcal C$.
Definition 2   Let $\ \emptyset\ne A\subseteq X,\ $ and
$\ a\in X\setminus A.\ $ Then the shadow $\ S(a\ A)\ $ of $\ A\ $
from $\ a\ $ is defined as follows:
$$ S(a\ A)\,\ :=\,\ \left\{x\in X:\ \forall_{D\in\mathcal C}\ \left(
\{a\,\ x\}\subseteq D\ \Rightarrow\ A\cap D\ne\emptyset\right)\right\} $$
There are $4$ axioms:


*

*$ \forall_{x\in X}\,\ \{x\}\in\mathcal C $

*$ \forall_{\mathcal D\subseteq\mathcal C}
      \,\ \bigcap \mathcal D\in\mathcal C\,\ $ (hence $\ X\in\mathcal C$)

*$ \forall_{A\subseteq X}\ ((\forall_{x\ y\in A}
     \exists_{D\in\mathcal C}\ \{x\ y\}\subseteq D)\ \Rightarrow
       \ A\in\mathcal C) $

*$ \forall_{A\in\mathcal C\setminus\{\emptyset\}}
     \forall_{a\in X\setminus A}\,\ S(a\ A)\in\mathcal C $


THEOREM
$$ \forall_{A\ B\in\mathcal C\setminus\{\emptyset\}}
        \,\ ((A\cap B=\emptyset)
\ \ \Rightarrow\ \ \exists_{G\ H\in\mathcal C}
\ (A\subseteq G\ \mbox{and}\ B\subseteq H\ \mbox{and}\ G\cap H=\emptyset)
      \,)  $$

Remark 1  
  The convex sets $\ G\ $ and $\ H\ $ from the theorem are
  half-spaces which separate disjoint convex sets $\ A\ $ and $\ B,\ $
  i.e. every disjoint pair of non-empty convex sets can be separated by half-spaces.
Remark 2   The above axioms admit an equivalent elementary version which does not use the notion of sets but only a kind of an in betweenness relation.

