# Thick Families of Geodesics and Differentiation

## TL;DR

This paper explores the relationship between thick families of geodesics in metric spaces and Lipschitz differentiability, providing new nonembeddability results and conditions for RNP differentiability in non-RNP Banach spaces.

## Contribution

It establishes that metric spaces with thick families of geodesics contain subsets with weakened RNP differentiability and constructs subsets with true RNP differentiability in non-RNP Banach spaces.

## Key findings

- Spaces with thick geodesic families contain subsets with weakened RNP differentiability.
- In non-RNP Banach spaces, such subsets can satisfy true RNP differentiability.
- New nonembeddability results for metric spaces into RNP Banach spaces.

## Abstract

The differentiation theory of Lipschitz functions taking values in a Banach space with the Radon-Nikod\'ym property (RNP), originally developed by Cheeger-Kleiner, has proven to be a powerful tool to prove non-biLipschitz embeddability of metric spaces into these Banach spaces. Important examples of metric spaces to which this theory applies include nonabelian Carnot groups and Laakso spaces. In search of a metric characterization of the RNP, Ostrovskii found another class of spaces that do not biLipschitz embed into RNP spaces, namely spaces containing thick families of geodesics. Our first result is that any metric space containing a thick family of geodesics also contains a subset and a probability measure on that subset which satisfies a weakened form of RNP Lipschitz differentiability. A corollary is a new nonembeddability result. Our second main result is that, if the metric space is a nonRNP Banach space, a subset consisting of a thick family of geodesics can be constructed to satisfy true RNP differentiability. An intriguing question is whether this differentiation criterion, or some weakened form of it such as the one we prove in the first result, actually characterizes general metric spaces non-biLipschitz embeddable into RNP Banach spaces.