Let $X$ be a real Hilbert space, and let $K$ be a closed convex cone.
The dual cone is defined by $K^*=\{x^*\in X \mid (\forall k\in K)\, \langle x^*,k\rangle \geq 0 \}$.
I am looking for some interesting examples of $K$ and $K^*$, especially when $X$ is infinite-dimensional.
For example,
- $X=\ell_2$ and $K=\ell_2^+ = K^*$
- $X=L_2[0,1]$ and $K=L_2^+[0,1]=K^*$
Note that these examples satisfy $K-K=X$. I wonder if there are some concrete examples where
$$K-K\neq X\quad \text{but}\quad \overline{K-K}=X$$
and $K^*$ is actually known?
Any examples/comments/references would be greatly appreciated.