Differentiating maps into<i>L</i>¹, and the geometry of<i>BV</i>functions

Abstract

This is one of a series of papers examining the interplay between differentiation theory for Lipschitz maps X ! V and bi-Lipschitz nonembeddability, where X is a metric measure space and V is a Banach space. Here, we consider the case V D L 1 , where differentiability fails. We establish another kind of differentiability for certain X , including n and , the Heisenberg group with its Carnot-Carathodory metric. It follows that does not bi-Lipschitz embed into L 1 , as conjectured by J. Lee and A. Naor. When combined with their work, this provides a natural counterexample to the Goemans-Linial conjecture in theoretical computer science; the first such counterexample was found by .