Separating hyperplane theorem for non-disjoint sets

[economicsnerd]

Consider the sets $X:= \{x\in\mathbb R^2: \enspace ||x-(-1,0)||_2 \leq 1\}$ (a ball) and $Y:=co\{(0,-1), (0,1), (1,0)\}$ (a triangle).

Both $X$ and $Y$ are compact and convex, but they aren't disjoint: $X\cap Y = \{(0,0)\}$. Since they aren't disjoint, the most common separating hyperplane/Hahn-Banach theorems don't directly apply. However, the y-axis is a hyperplane which separates them in a weak sense.

Can anybody point me to source for a separating hyperplane theorem that covers this example? Ideally, I'm looking for something that isn't restricted to finite dimensions.

Thanks!

 

[micromass]

Consider the sets $X:= \{x\in\mathbb R^2: \enspace ||x-(-1,0)||_2 \leq 1\}$ (a ball) and $Y:=co\{(0,-1), (0,1), (1,0)\}$ (a triangle).

Both $X$ and $Y$ are compact and convex, but they aren't disjoint: $X\cap Y = \{(0,0)\}$. Since they aren't disjoint, the most common separating hyperplane/Hahn-Banach theorems don't directly apply. However, the y-axis is a hyperplane which separates them in a weak sense.

Can anybody point me to source for a separating hyperplane theorem that covers this example? Ideally, I'm looking for something that isn't restricted to finite dimensions.

Thanks!

Can't you just apply Hahn-Banach to the interiors?

 

[economicsnerd]

Oh jeez. Now I'm embarrassed.

Thank you, micromass!

Can we strike this one from the records? :P