Separation of convex sets in finite dimensionale space

Question

I’m trying to proof that if $A$ and $B$ are two convex sets in a finite dimensional vector space $V$, with $A\cap B= \emptyset $ , than there exists an hyperplane that separates $A$ and $B$, but I’m not understanding how to proceed. Does someone have any ideas?

Answer 1

You should start with the case that $A$ and $B$ are both compact -- this makes the proof much easier. Simply take $a \in A$ and $b \in B$ such that $\lVert a - b \rVert$ is minimized, and then show that the hyperplane bisecting the line segment between $a$ and $b$ works.

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For the general case, use a compact exhaustion argument. Write $A = \bigcup_{i \in \mathbb{N}} A_i$ and $B = \bigcup_{i \in \mathbb{N}} B_i$ as increasing unions of compact convex sets. For each $i$, pick a hyperplane $H_i = \{x \in V : x \cdot v_i = c_i\}$ separating $A_i$ and $B_i$ (here $\cdot$ denotes some fixed inner product on $V$, $c_i$ is some real number, and $v_i$ is some unit vector). Then use compactness of the unit sphere in $V$ to choose a subsequence where the $v_i$'s converge. Then show that there is some $c$ such that $H := \{x \in V : x \cdot \lim_i v_i = c\}$ separates $A$ from $B$.