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Boyd's book on convex optimization makes the following claim about cones and their duals-

If $K$ is a cone, and its closure is pointed, then $K^*$ has nonempty interior.

Any hints on how to prove this? From sketching a few pictures in $\mathbb{R}^2$ it seems like a geometric approach would be useful- if the closure is pointed, the "opening angle" of the cone should be less than 180 degrees, which ought to produce a nonempty interior for the dual. That was my thinking, anyway, but I'm struggling to make the line of reasoning tangible.

btimar
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1 Answers1

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I am assuming that $K \subset \mathbb{R}^n$.

Suppose $K^*$ has an empty interior. Then $K^* \subset L$ where $L$ is a linear subspace of dimension $ \le n-1$.

Since $\overline{K} = K^{**}$, we see that $L^* = L^{\bot} \subset \overline{K}$ and so $\overline{K}$ is not pointed.

copper.hat
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