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Proceedings of the American Mathematical Society

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The logarithmic center of a planar region

Author: Douglas Hensley
Journal: Proc. Amer. Math. Soc. 57 (1976), 266-270
MSC: Primary 31A10
MathSciNet review: 0407291
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Abstract: Given a bounded region $ S$ in the complex plane, let $ f(\beta ) = {\smallint _S}\log \vert z - \beta \vert d$ area for $ \beta $ any complex number. A logarithmic center of $ S$ is an $ \alpha $ which minimizes $ f(\beta )$. When is $ \alpha $ unique?

Conjecture. If $ S$ is convex then $ \alpha $ is unique.

Theorem. If $ S$ is convex and symmetric about some line, then $ \alpha $ is unique.

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

  • [1] Douglas Hensley, An asymptotic inequality concerning primes in contours for the case of quadratic number fields, Acta Arith. 28 (1975/76), no. 1, 69–79. MR 0382196
  • [2] I. Kubilyus, The distribution of Gaussian primes in sectors and contours, Leningrad. Gos. Univ. Uč. Zap. Ser. Mat. Nauk 137(19) (1950), 40–52 (Russian). MR 0079610
  • [3] Marston Morse and Stewart S. Cairns, Critical point theory in global analysis and differential topology: An introduction, Pure and Applied Mathematics, Vol. 33, Academic Press, New York-London, 1969. MR 0245046
  • [4] Tibor Radó, On the problem of Plateau. Subharmonic functions, Springer-Verlag, New York-Heidelberg, 1971. Reprint. MR 0344979

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Article copyright: © Copyright 1976 American Mathematical Society