The question is:
Suppose P is a cone in E and C is a compact and convex set of a Euclidean space $Y$. $K$ is a cone in $E \times Y$. Prove: $$ (K\cap (P \times C))^{\circ} = (K\cap (P \times C^{\circ \circ}))^\circ $$ where $C^\circ=\{y|y^T x≤1$ for all $x∈C\}$.
We know that if $C$ is compact and convex, then $C^{\circ \circ}=U_{λ∈[0,1]}λC$.
I have a counterexample:
Let $C=\{1\}$, and $P=R_+$, and $K=R_+ \times \{0\}$.
Then $K\cap (P \times C)=\emptyset$, but $K\cap (P \times C^{\circ \circ})=K$.
Therefore,$(K\cap (P \times C))^{\circ}=R^2$. $(K\cap (P \times C^{\circ \circ}))^\circ=-K$.
Is my counterexample wrong? Anyone there can help? Thanks.
I think it should be:
$(K\cap (P \times C))^{\circ}=(K\cap (P \times A))^\circ$.
where $A=U_{0<\lambda \le 1} \lambda C$