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 r 1 , … , r n in the image of g , there is another u v -geodesic g ˜ ∈ T satisfying the conditions: 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 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 α .