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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.

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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:

  1. $ \forall_{x\in X}\,\ \{x\}\in\mathcal C $
  2. $ \forall_{\mathcal D\subseteq\mathcal C} \,\ \bigcap \mathcal D\in\mathcal C\,\ $ (hence $\ X\in\mathcal C$)
  3. $ \forall_{A\subseteq X}\ ((\forall_{x\ y\in A} \exists_{D\in\mathcal C}\ \{x\ y\}\subseteq D)\ \Rightarrow \ A\in\mathcal C) $
  4. $ \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.

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    $\begingroup$ Thank you very much for your answer(+1). Actually in separation for LCTV the important thing is separating disjoint convex sets through CONTINUOUS linear function, that leads us to get good result . Other wise any two convex set in a vector space can be separated by two disjoint maximal convex set which make a partition for X, so they are actually two sides of a half- space. But I am looking for a continuous separation ! $\endgroup$ – Red shoes Jun 24 '17 at 8:33
  • $\begingroup$ Ashkan, very true. Nevertheless, starting with two disjoint non-empty convex sets which have non-empty interiors (plus perhaps some extra conditions) one could try to prove that the closures of the separating half-spaces intersect in a codimension 1 hyperplane. You could add additional specific conjectures along this line which would expand on your question providing more detailed "subquestions". Etc. (Thank you for your generous +1). $\endgroup$ – Wlod AA Jun 24 '17 at 14:35

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