For a subset $A\subset V$ of a vector space over $\mathbb{R}$, let
$\mbox{conv}(A) := \left\{ \sum_{i=1}^n a_i x_i\, \middle| \,x_i\in A, a_i \ge 0\text{ with } \sum_{i=1}^n a_i = 1 \right\}$.
I want to show that if for all $x,y\in A$ and $\alpha\in[0,1]$ the vector $(1-\alpha)x + \alpha y $ is an element of $A$, then $A = \mbox{conv}(A)$.
Is there an elegant solution? Thank you for your help!