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Lipschitz representations of subsets of the cube

Author: Shahar Mendelson
Journal: Proc. Amer. Math. Soc. 135 (2007), 1455-1463
MSC (2000): Primary 46B07, 60D05
Published electronically: November 14, 2006
MathSciNet review: 2276655
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Abstract: We show that for any class of uniformly bounded functions $ H$ with a reasonable combinatorial dimension, the vast majority of small subsets of the $ n$-dimensional combinatorial cube cannot be represented as a Lipschitz image of a subset of $ H$, unless the Lipschitz constant is very large. We apply this result to the case when $ H$ consists of linear functionals of norm at most one on a Hilbert space.

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Shahar Mendelson
Affiliation: Centre for Mathematics and its Applications, The Australian National University, Canberra, ACT 0200, Australia

Received by editor(s): April 29, 2005
Received by editor(s) in revised form: December 20, 2005
Published electronically: November 14, 2006
Additional Notes: The author was supported in part by an Australian Research council Discovery grant.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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