I want to show the following claim:
$U \subset \mathbb{R}^n$ is open and convex, and $f$ is a convex function from $U$ to $\mathbb{R}$. For fix $x_0 \in U$ there is a $v \in \mathbb{R}^n$ such that the following holds:
$f(x)\geq f(x_0)+\sum_{i=0}^{n}v_i (x-x_0)_i$.
I was thinking of an application of the following form of separation theorem on normed spaces $X$:
For $V$ (open) and $W$ disjoint convex sets there is a $x^* \in X^*$ such that $Re(x^*(v)) < Re(x^*(w))$ $\forall v\in V$ and $w\in W$;
to the set $V:=\{(t,x)$ so that $x\in U $ and $ t> f(x)\}$.
Do you have any suggestions how this works out?