Abstract
This is one of a series of papers examining the interplay between differentiation theory for Lipschitz maps $X\to V$ 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 $\mathbb{R}^n$ and $\mathbb{H}$, the Heisenberg group with its Carnot-Carathéodory metric. It follows that $\mathbb{H}$ 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].
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[semmes] S. Semmes, "On the nonexistence of bi-Lipschitz parameterizations and geometric problems about $A_\infty$-weights," Rev. Mat. Iberoamericana, vol. 12, iss. 2, pp. 337-410, 1996.
@article {semmes, MRKEY = {1402671},
AUTHOR = {Semmes, Stephen},
TITLE = {On the nonexistence of bi-{L}ipschitz parameterizations and geometric problems about {$A\sb \infty$}-weights},
JOURNAL = {Rev. Mat. Iberoamericana},
FJOURNAL = {Revista Matemática Iberoamericana},
VOLUME = {12},
YEAR = {1996},
NUMBER = {2},
PAGES = {337--410},
ISSN = {0213-2230},
MRCLASS = {30C65 (42B20)},
MRNUMBER = {97e:30040},
MRREVIEWER = {Guy David},
} -
[ziemer] W. P. Ziemer, Weakly Differentiable Functions, New York: Springer-Verlag, 1989.
@book {ziemer, MRKEY = {1014685},
AUTHOR = {Ziemer, W P.},
TITLE = {Weakly Differentiable Functions},
SERIES = {Grad. Texts Math.},
NUMBER = {120},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1989},
PAGES = {xvi+308},
ISBN = {0-387-97017-7},
MRCLASS = {46E35},
MRNUMBER = {91e:46046},
MRREVIEWER = {V. M. Gol{\cprime}dshte{\u\i}n},
ZBLNUMBER = {0692.46022},
}
Full Article