Differentiating maps into L1, and the geometry of BV functions

By Jeff Cheeger, Bruce Kleiner. 2006

114 citations recorded by the source corpus.

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-Caratheodory 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 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 L 1 and functions of bounded variation, which permits us to exploit results on the structure of BV functions on the Heisenberg group [FSSC01].

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/971b1236dae28d1452909e447ace1b4fbb80c365

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Characterizations of intrinsic rectifiability in Heisenberg groups (2010, background): “See also, for a different application, the recent [10].”
Rectifiability of Sets of Finite Perimeter in Carnot Groups: Existence of a Tangent Hyperplane (2008, result): “measure (X1 E,...,Xm E) plays the role of surface measure associated to d. Our interest in this topic was also motivated by the recent papers [7], [ 8 ], where sets of finite perimeter in Carnot groups (and in particular in the Heisenberg groups) are used to study a new notion of dierentiability for”
On the differentiability of Lispschitz maps from metric measure spaces to Banach spaces (2006, methodology): “Nonetheless, as we will show elsewhere, by using a different notion of differentiation, the Heisenberg group with its CarnotCaratheodory metric does not admit a bi-Lipschitz embedding into L(1); see [CK06a].”
Lp compression, traveling salesmen, and stable walks (2009, background): “Note that not every group of polynomial growth H admits a bi-Lipschitz embedding into L1, as shown by Cheeger and Kleiner [17] when H is the discrete Heisenberg group, i.”