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The convexity of a domain and the superharmonicity of the signed distance function

Authors: D. H. Armitage and Ü. Kuran
Journal: Proc. Amer. Math. Soc. 93 (1985), 598-600
MSC: Primary 31B05; Secondary 52A20
MathSciNet review: 776186
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Abstract: Let $ D$ be a domain in $ {{\mathbf{R}}^N}$ with nonempty boundary $ \partial D$ and let $ u$ be the signed distance function from $ \partial D$, i.e. $ u = \pm $ dist according as we are in or outside $ \overline D $. We prove that, for any $ N \geqslant 2,u$ is superharmonic in $ {{\mathbf{R}}^N}$ if and only if $ D$ is convex. When $ N = 2$, this criterion requires the superharmonicity of $ u$ in $ D$ only.

References [Enhancements On Off] (What's this?)

  • [1] W. H. J. Fuchs, Topics in the theory of functions of one complex variable, Manuscript prepared with the collaboration of Alan Schumitsky. Van Nostrand Mathematical Studies, No. 12, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0220902
  • [2] Frederick A. Valentine, Convex sets, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Toronto-London, 1964. MR 0170264

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Keywords: Superharmonic function, distance, convex set
Article copyright: © Copyright 1985 American Mathematical Society

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