Conic hull of a proper function

Question

Suppose $f$ is a proper function pn $\mathbb{R}^{n}$with $f(0)>0$.Now consider $$ g(x) = \text{inf}\{t: (t,x) \in \text{cl(cone(epi(}f)))\} $$ Can I always say that $\exists y \in \mathbb{R}^{n} : g(y) = f(y)$? How do I characterize such a $y$?

Answer 1

Take $f(x) = 1$, then $\operatorname{epi} f = \{(\mu,x) | 1\le \mu \}$. Then $\overline{\operatorname{cone}(\operatorname{epi} f)} = \{(\mu,x) | 0 \le \mu \}$, and so $g(x) = 0$.