Lipschitz representations of subsets of the cube

Shahar Mendelson

doi:10.1090/S0002-9939-06-08620-5

Lipschitz representations of subsets of the cube

Shahar Mendelson^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

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.

Original language	English
Pages (from-to)	1455-1463
Number of pages	9
Journal	Proceedings of the American Mathematical Society
Volume	135
Issue number	5
DOIs	https://doi.org/10.1090/S0002-9939-06-08620-5
Publication status	Published - May 2007

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AB - 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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Lipschitz representations of subsets of the cube

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