[Banach] Abstract of a paper by Jeff Cheeger and Bruce Kleiner

[Dale Alspach]

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

or by email in unzipped form by transmitting an empty message with
subject line

	 uget 0611954


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	 get 0611954

 to: math at arXiv.org.