  Let $V$ be a finite nonempty set. A transit function is a map $R:V\times
V\rightarrow 2^V$ such that $R(u,u)=\{u\}$, $R(u,v)=R(v,u)$ and $u\in R(u,v)$
hold for every $u,v\in V$. A set $K\subseteq V$ is $R$-convex if $R(u,v)\subset
K$ for every $u,v\in K$ and all $R$-convex subsets of $V$ form a convexity
$\mathcal{C}_R$. We consider Minkowski-Krein-Milman property that every
$R$-convex set $K$ in a convexity $\mathcal{C}_R$ is the convex hull of the set
of extreme points of $K$ from axiomatic point of view and present a
characterization of it. Later we consider several well-known transit functions
on graphs and present the use of the mentioned characterizations on them.
