This is an announcement for the paper "Differentiating maps into L^1 and the geometry of BV functions" by Jeff Cheeger and Bruce Kleiner.
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.
Archive classification: Metric Geometry; Differential Geometry; Functional Analysis; Group
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/math.MG/0611954
or
http://arXiv.org/abs/math.MG/0611954
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