I am studying convex analysis especially the structure of closed convex sets. I need a clarification on something that sounds quite easy but I can't put my fingers on it.
Let $E$ be a normed VS of a finite demension. We consider in the augmented vector space $\hat{E}=E\oplus\mathbb{R}$ the convex $\hat{C}=C\times\{1\}$ (obtained by translation from the convex $C=C\times\{0\}$), and the convex cone $\Gamma(C)=cone\hat{C}=\bigcup_{\lambda\geq0}\lambda\hat{C}$.
It's obvious to see that $\Gamma(C)\subset E\times\mathbb{R}_+$, but here are my questions:\
- What about the closure $\overline{\Gamma(C)}$? Is $\overline{\Gamma(C)}\subset E\times\mathbb{R}_+$ too?
- If so, why?
- Does $\overline{\Gamma(C)}$ need to be salient to verify that?
We define a salient cone as:
$\Gamma$ is salient if $\Gamma\cap-\Gamma\subseteq\{0\}$.
While trying to see that I thought of using the fact that the adherence of a convex cone is a convex cone too (and I added a condition that $\overline{\Gamma(C)}$ is salient), so if $\overline{\Gamma(C)}$ contains an element of $E\times]-\infty,0[$, then the conic combination with any element of $\overline{\Gamma(C)}$ would still be in $\overline{\Gamma(C)}$. Therefore, the intersection of $\overline{\Gamma(C)}$ and $-\overline{\Gamma(C)}$ wouldn't be contained in $\{0\}$.
Is this correct?