Consider an indicator function $I$ defined as $$ I\left( {x,y} \right): = \begin{cases} 0,&\left( {x,y} \right) \in [0,+\infty) \times [0,+\infty)\\ + \infty, & \text{otherwise} \\ \end{cases} $$ I want to find the "recession cone of its epigraph"; namely, I want to find $rc (epi (I) )$.
Here is my thinking: $epi(I)$ is the set $\{( (x,y), \xi) \in \mathbb{R}^{2+1}: x\geq 0, y \geq 0\}$ which gives me the same $rc(epic(I))$. That is, $$ rc(epi (I) ) = \{(x,y) \times \mathbb{R}_+ : x \geq 0, y \geq 0\} $$ Is my thinking correct?
Yes, the epigraph of $I$ is a cone. Hence, it equals its recession cone.