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

Maleva, Olga

doi:10.1090/S0002-9939-04-07657-9

Components of level sets of uniform co-Lipschitz functions on the plane
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Proc. Amer. Math. Soc. 133 (2005), 841-850 Request permission

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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