Differentiating maps into L^1 and the geometry of BV functions

TL;DR

This paper introduces a new form of differentiability for Lipschitz maps into L^1, demonstrating that the Heisenberg group cannot bi-Lipschitz embed into L^1, thus providing a counterexample to a conjecture in theoretical CS.

Contribution

It establishes a novel differentiability concept for certain metric spaces into L^1 and applies it to prove non-embeddability of the Heisenberg group into L^1.

Findings

Heisenberg group does not bi-Lipschitz embed into L^1

New connection between Lipschitz maps and BV functions

Counterexample to the Goemans-Linial conjecture

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=L^1 where differentiability fails. We establish another kind of differentiability for certain X, including R^n and H, the Heisenberg group with its Carnot-Cartheodory metric. It follows that H 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 counter example to the Goemans-Linial conjecture in theoretical computer science; the first such counterexample was found by Khot-Vishnoi. A key ingredient in the proof of our main theorem is a new connection between Lipschitz maps to L^1 and functions of bounded variation, which permits us to exploit recent work on the structure of BV functions on the Heisenberg group.