Radon-Nikodým property and thick families of geodesics

Banach spaces without the Radon-Nikodým 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 thick if there is α>0 such that for every g∈T and for any finite collection of points r1,…,rn in the image of g, there is another uv-geodesic g˜∈T satisfying the conditions: g˜ also passes through r1,…,rn, and, possibly, has some more common points with g. On the other hand, there is a finite collection of common points of g and g˜ which contains r1,…,rn and is such that the sum of maximal deviations of the geodesics between these common points is at least α.