Let $S$ be a closed unbounded convex set of $\mathbb{R}^2$ and assume that the asymptotic cone $S_{\infty}$ of $S$ has no empty interior. (It is not a line or a half line) Of course I know that I can find $z \in \mathbb{R}^2$ such that $z+S_{\infty} \subset S.$ I would ask if the converse is also true. Does there exist some $z\in \mathbb{R}^2$ such that $S \subset z + S_{\infty} ?$
On a picture it is obviously false if the asymptotic cone has empty interior, but if not it seems pretty intuitive.
Try $S = \{(x,y): x \ge 0, -x - \sqrt{x} \le y \le x + \sqrt{x}\}$, for which the asymptotic cone is $\{(x,y): |y| \le x\}$.