I am doing some work with Newton Polytopes and I need something of this style:
Given $v_1,\dots,v_n\in \mathbb{R}^n$ we have
$$\text{conv}(v_1,\dots v_n)=\{v\in \mathbb{R}^n\mid \min_{i=1,\dots n}\{x\cdot v_i\}\leq x\cdot v \;\;\forall \, x\in \mathbb{R}^n\}$$
As the condition defining the set in the left is true for the $v_i$ and is closed under convex combinations then $\subseteq$ is easy. My problem is to show that
$$\min_{i=1,\dots n}\{x\cdot v_i\}\leq x\cdot v \;\;\forall \, x\in \mathbb{R}^n\implies v\in \text{conv}(v_1,\dots v_n)$$
This is trivial for $n=1$ but I can't see it for all $n$.
Any help is appreciated.