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ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-1985-0776186-8
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.


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DOI: https://doi.org/10.1090/S0002-9939-1985-0776186-8
Keywords: Superharmonic function, distance, convex set
Article copyright: © Copyright 1985 American Mathematical Society

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