Strict inclusion for recession cone of closure of a convex set

Question

Let $C$ be a nonempty closed convex subset of $\mathbb{R}^n$. The recession cone of $C$ is given by $$R_C=\left\lbrace d\in\mathbb{R}^n:x+td\in C, \forall t>0, \forall x\in C\right\rbrace.$$ It is shown that $R_C$ is a closed, convex cone in $\mathbb{R}^n$.

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Let $C$ be a convex subset and $\overline{C}$ be the closure of $C$. We can show that $$R_C\subset R_{\overline{C}}\qquad \text{and}\qquad \overline{R_{C}}\subset R_{\overline{C}}.$$ I want to find examples where these inclusions are strict. I tried some $C$ but found no subset that satisfies. Please help me with some examples. Thank you very much.

P/s: in some books, I see that the closeness of $C$ in the recession cone definition is unnecessary.

Answer 1

The following set taken from here fulfills the inclusions strict: $$ C = \left\lbrace(x,y) \mid 0 \leq x < 1, y \geq 1\right\rbrace \cup \left\lbrace(x,y) \mid 0 \leq x \leq 1, 0 \leq y \leq 1\right\rbrace. $$ Then $R_C = \{0\}$ but $R_{\overline{C}}$ is the cone generated by the direction $(0,1)$.