The convexity of a domain and the superharmonicity of the signed distance function
HTML articles powered by AMS MathViewer
- by D. H. Armitage and Ü. Kuran PDF
- Proc. Amer. Math. Soc. 93 (1985), 598-600 Request permission
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
- W. H. J. Fuchs, Topics in the theory of functions of one complex variable, Van Nostrand Mathematical Studies, No. 12, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. Manuscript prepared with the collaboration of Alan Schumitsky. MR 0220902
- Frederick A. Valentine, Convex sets, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Toronto-London, 1964. MR 0170264
Additional Information
- © Copyright 1985 American Mathematical Society
- 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