This is an announcement for the paper "Radon-Nikod'ym property and thick families of geodesics" by Mikhail I. Ostrovskii.

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 $r_1,\dots,r_n$ in the image of $g$, there is another $uv$-geodesic $\widetilde g\in T$ satisfying the conditions: $\widetilde g$ also passes through $r_1,\dots,r_n$, 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 $r_1,\dots,r_n$ and is such that the sum of maximal deviations of the geodesics between these common points is at least $\alpha$.

Archive classification: math.FA math.MG

Mathematics Subject Classification: 46B22, 46B85, 54E35

Submitted from: ostrovsm@stjohns.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1306.5807

or

http://arXiv.org/abs/1306.5807