Does the dual of this cone have a non-empty interior?

Question

Let $(X, X')$ be a dual pair, and let $K \subset X$.

Let $K' = \{x' \in X': \langle x, x' \rangle >0, \forall x \in K \}$.

Finally, let $D_K = \{x \in X: \langle x,x'\rangle > 0, \forall x' \in K'\}.$

It's easy to see that $D_K$ is convex, but

Does $D_K$ have a non-empty interior in the weak topology?

I have been able to observe the following. If $K$ is compact, then $K'$ is open in the weak* topology. If $K$ is also convex, then $K = D_K$. I'm not sure what to do about general $K$ though. I suppose it would also help me to know the answer to

Does $K'$ have a non-empty interior in the weak* topology?

Answer 1

Not always

For example if $X=X'=\mathbb R$ the weak topology is the same as the Euclidean topology. If $K$ is compact without interior then, as you said, we have $D_K=K$ does not have interior.