Radon-Nikod\'ym property and thick families of geodesics

Banach spaces without the Radon-Nikod\'ym property are characterized as spaces containing bilipschitz images of thick families of geodesics defined as follows. A family $T$ of geodesics joining points $u$ and $v$ in a metric space is called {\it thick} if there is $\alpha>0$ such that for every $g\in T$ and for any finite collection of points $r_1,...,r_n$ in the image of $g$, there is another $uv$-geodesic $\widetilde g\in T$ satisfying the conditions: $\widetilde g$ also passes through $r_1,...,r_n$, and, possibly, has some more common points with $g$. On the other hand, there is a finite collection of common points of $g$ and $\widetilde g$ which contains $r_1,...,r_n$ and is such that the sum of maximal deviations of the geodesics between these common points is at least $\alpha$.