Radon-Nikodým property and thick families of geodesics

Abstract

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 {\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 $\tilde{g}\in T$ satisfying the conditions: $\tilde{g}$ 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 $\tilde{g}$ 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 $\alpha$.

References in corpus (1)

• On metric characterizations of the Radon-Nikodým and related properties of Banach spaces

Cited by in corpus (2)

• Purely 1-unrectifiable metric spaces and locally flat Lipschitz functions
• Metric spaces nonembeddable into Banach spaces with the Radon-Nikodým property and thick families of geodesics