Thick families of geodesics and differentiation

The differentiation theory of Lipschitz functions on metric spaces taking values in a Banach space with the Radon-Nikodým property (RNP), originally developed by Cheeger—Kleiner, has proven to be a powerful tool to prove non-bi-Lipschitz 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 bi-Lipschitz 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 non-RNP 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-bi-Lipschitz embeddable into RNP Banach spaces.