Relative interior and epigraph of convex set

Question

Consider a Banach space $X$ and a non empty convex set $C \subseteq X$ and let $f : C \to \mathbb{R}$ be a convex functional.

1. Does $C$ have a non-empty relative interior? If no, what necessary and sufficient conditions should $C$ satisfy for it to have a non-empty relative interior?

2. Does the epigraph of $f$ over $C$ defined as $[f,C] := \{ (r,x) \in \mathbb{R}\times X : r \ge f(x) \}$ have a non-empty relative interior? If it does not necessarily have a relative interior, what necessary and sufficient conditions should $f$ or $C$ satisfy for it to have a relative interior? If it exists, is the relative interior precisely equal to the set $\{ (r,x) \in \mathbb{R}\times X : r > f(x) \}$, and if yes, how does one prove it?