use of Hahn-Banach theorem to prove the existence of a function

Question

Let $E$ be an $\Bbb R$-vector space and $g:E\to \Bbb R$ a convex function s.t. $g(0)\ge 0$. let $g_1:E\to \Bbb R$ s.t. $g_1(x)=\inf_{t>0}\frac{1}{t}g(tx)$.

Prove that there is a function $f:E\to \Bbb R$ s.t. $f(x)\le g(x)$ and $f(x)\le g_1(x), \forall x\in E$.

It feels like the conditions of Hahn-Banach since g is subadditive and positively homogeneous but I don't see which linear function I can use to apply Hahn-Banach.

Thank you for your help.

Answer 1

Since $0\in int\ dom\ g$, the subdifferential $\partial g(0)$ is non-empty. Take $w\in \partial g(0)$. Then $\langle w,x\rangle \le g(x)-g(0)\le g(x)$ for all $x$ as $g(0)\ge0$.

Then $$ \frac 1t g(tx) \ge \frac1t \langle w,tx\rangle = \langle w,x\rangle. $$ which proves $g_1(x) \ge \langle w,x\rangle$. Hence, the claim is true with $f(x) = \langle w,x\rangle$.