Convex Geometries yielded by Transit Functions

Let $V$ be a finite nonempty set. A transit function is a map $R:V\times V\to 2^V$ 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.