I do not know a definitive "weakest" condition and I doubt there is one. Many results in the realm of Hahn-Banach do the trick, i.e. there is the general result for $A,B$ convex and $A$ open (both open giving strict separation) and there is also Eidelheit's theorem saying that you can separate a point from a closed convex set (or a compact convex from a closed convex one). The latter one also holds for convex $A,B$ such that the interior of $A$ is non-empty and does not intersect $B$.
I do not know a definitive "weakest" condition and I doubt there is one. Many results in the realm of Hahn-Banach do the trick, i.e. there is the general result for $A,B$ convex and $A$ open (both open giving strict separation) and there is also Eidelheit's theorem saying that you can separate a point from a closed convex set (or a compact from a closed one). The latter one also holds for convex $A,B$ such that the interior of $A$ is non-empty and does not intersect $B$.
I do not know a definitive "weakest" condition and I doubt there is one. Many results in the realm of Hahn-Banach do the trick, i.e. there is the general result for $A,B$ convex and $A$ open (both open giving strict separation) and there is also Eidelheit's theorem saying that you can separate a point from a closed convex set (or a compact convex from a closed convex one). The latter one also holds for convex $A,B$ such that the interior of $A$ is non-empty and does not intersect $B$.
I do not know a definitive "weakest" condition and I doubt there is one. Many results in the realm of Hahn-Banach do the trick, i.e. there is the general result for $A,B$ convex and $A$ open (both open giving strict separation) and there is also Eidelheit's theorem saying that you can separate a point from a closed convex set (or a compact from a closed one). The latter one also holds for convex $A,B$ such that the interior of $A$ is non-empty and does not intersect $B$.