I am studying convex analysis at home for self-learning. I focus especially on the structure of closed convex sets. I came across a weird definition of an asymptotic cone in a French (Sorbonne) university handbook:
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 $\Gamma(C)=cone\hat{C}=\bigcup_{\lambda\geq0}\lambda\hat{C}$.
We call an asymptotic cone of a convex set $C$ the closed convex cone $C_{as}$ such that: $$\overline{\Gamma(C)}\cap(E\times\{0\})=C_{as}\times\{0\}$$
My question is: Isn't this $C_{as}=\{0\}$? Wouldn't it shrink to a point? This doesn't seem like it's the case because later on the handbook they will define lines, half-lines and directions in this cone. Does any one have any intuition on that? Does it even has any relationship with the defintion of asymptotic (recession) cone on Wikipedia?