October 13, 2004
This notion is due to K. Ball (1992), who showed its importance for Lipshitz extensions. E.g $L^2$ has uniform Markov type 2, but expander graphs and hypercubes do not. Saying this quantitatively yields "robust" lower bounds on distortion of embeddings, that are sharp in many cases. Recently, we showed that $L^p$ for fixed finite $p>2$, trees and Gromov hyperbolic graphs all have uniform Markov type $2$; it is open whether planar graphs have this property.
(joint work with A. Naor, S. Sheffield and O. Schramm).