Differentiating maps into L¹, and the geometry of BV functions

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 = L¹, where differentiability fails. We establish another kind of differentiability for certain X, including R{double-struck}ⁿ and H{double-struck}, the Heisenberg group with its Carnot-Carathéodory metric. It follows that H{double-struck} does not bi-Lipschitz embed into L¹, 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 Khot-Vishnoi [KV05]. A key ingredient in the proof of our main theorem is a new connection between Lipschitz maps to L¹ and functions of bounded variation, which permits us to exploit results on the structure of BV functions on the Heisenberg group [FSSC01].