Let's denote by $\cal{F}$ the family of all affine functions $f: \mathbb{R}^n \rightarrow\mathbb{R}$. Let $A\subset \mathbb{R}^n$.
What is a connection between the following definitions of convex hull of $A$:
$conv_1(A)=\{\sum_{i=1}^n \alpha_i a_i: \alpha_i \in [0,1], a_i \in A, \textrm{ for } i=1,...,n; \sum_{i=1}^n \alpha_i=1, n \in \mathbb{N} \},$
$conv_2(A)= \bigcap_{f \in \cal{F} } \{ x\in \mathbb{R}^n : |f(x)| \leq \sup_{y \in A} |f(y)| \}.$
Thanks.