Let $A,B$ be two sets. We say the hyperplane $\langle a,x\rangle =c$ separates $A,B$ if $A\subset H^-$ and $B\subset H^+$, that is $$ x\in A \implies \langle a,x\rangle \leq c\\ x\in B \implies \langle a,x \rangle \geq c $$ I am confused about the fact that both of these inequalities allow equality. If both of them are equal than can't the hyperplanes be touching at one point? This doesn't make sense to me unless we allow the half spaces to be closed (Which I guess we must).
I am wondering then, why the separating hyperplane theorem requires that $A\cap B =\emptyset$ though? Is it just that the separating hyperplane theorem only applies when the intersection is empty, but there can be a separating hyperplane when the sets touch at one point? (my interactions with separating hyperplanes up to this point was only through this theorem, and I did not/still don't fully understand it, so until now I had assumed that the sets must have an empty intersection, which seems to perhaps not be true in order to have a separating hyperplane, but which obviously must be true to apply the separating hyperplane theorem)
Also, in the case where there is equality, then the supporting hyperplane theorem will always be applicable as well, won't it (on both sets given they are convex), since the point the hyperplane passes through seems like it must be on the boundary of each of the sets.
Assume convex, non-empty sets.