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

TL;DR

This paper characterizes Banach spaces lacking the Radon-Nikodým property by their ability to contain bilipschitz images of thick families of geodesics, introducing a new geometric criterion involving geodesic deviations.

Contribution

It introduces the concept of thick families of geodesics and links their presence to the absence of the Radon-Nikodým property in Banach spaces.

Findings

Banach spaces without RNP contain bilipschitz images of thick geodesic families.

Thick families of geodesics are characterized by a uniform deviation property.

The geometric structure of geodesic families characterizes the RNP property.

Abstract

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 $\alpha$.