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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Components of level sets of uniform co-Lipschitz functions on the plane


Author: Olga Maleva
Journal: Proc. Amer. Math. Soc. 133 (2005), 841-850
MSC (2000): Primary 46B20
Published electronically: September 29, 2004
MathSciNet review: 2113935
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Abstract: Consider a co-Lipschitz uniformly continuous function $f$defined on the plane. Let $n(f)$ be the maximal number of components of its level set. In the present paper we settle a question of B. Randrianantoanina, concerning the dependence of $n(f)$ on the quantitative characteristics of the mapping. We prove that $n(f)$ is bounded from above by a simple function of the co-Lipschitz and the ``weak Lipschitz'' constants of $f$, and show that our estimate is sharp. We also prove additional properties of the level sets.


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Additional Information

Olga Maleva
Affiliation: Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel
Address at time of publication: Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom
Email: olga@math.ucl.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07657-9
PII: S 0002-9939(04)07657-9
Received by editor(s): November 5, 2002
Received by editor(s) in revised form: November 20, 2003
Published electronically: September 29, 2004
Additional Notes: The author was supported by the Israel Science Foundation.
Communicated by: David Preiss
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.