Minkowski sum of duals

Question

I'm really struggling to prove the following statement:

Let $\mathbb{E}$ be an Euclidean space, let $K,K_p,S\subseteq\mathbb{E}$ be a proper cone, a polyhedral cone and a subspace, respectively. If $\text{int}(K)\cap K_p\cap S\not=\varnothing$, then $K^\ast+K_p^\ast+S^\bot$ is closed.

I know that this result is essentially a sub-result of Corollary 20.1.1 from Rockafellar's book. However, I want to prove it separately from the other parts. Does anyone know anywhere that this result is shown or has any hints?

Answer 1

This result may be shown applying Theorem 20.2 to prove Lema 16.2 and Corollary 16.2.2 with the required condition. Then, one is able to prove Thm 16.4 with this condition and finally derive the desired result as done in Corollary 16.4.2.