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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A round ball uniquely minimizes gravitational potential energy
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by Frank Morgan PDF
Proc. Amer. Math. Soc. 133 (2005), 2733-2735

Abstract:

We give a proof following Carleman that among measurable bodies in $\mathbf {R}^3$ of mass $m_0$ and density at most 1, a round ball of unit density uniquely minimizes gravitational potential energy.
References
  • Wilhelm Blaschke, Eine isoperimetrische Eigenschaft des Kreises, Math. Z. 1 (1918), 52–57.
  • T. Carleman, Über eine isoperimetrische Aufgabe und ihre physikalischen Anwendungen, Math. Z. 3 (1919), 1–7.
  • S. Chandrasekhar, Ellipsoidal figures of equilibrium—an historical account, Comm. Pure Appl. Math. 20 (1967), 251–265. MR 213075, DOI 10.1002/cpa.3160200203
  • Frank Morgan, The perfect shape for a rotating rigid body, Math. Magazine 75 (February, 2002), 30–32.
  • H. Poincaré, Figures d’Equilibre d’une Masse, Paris, Gauthier-Villars, 1902.
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Additional Information
  • Frank Morgan
  • Affiliation: Department of Mathematics and Statistics, Williams College, Williamstown, Massachusetts 01267
  • Email: Frank.Morgan@williams.edu
  • Received by editor(s): January 14, 2002
  • Published electronically: April 12, 2005
  • Communicated by: Wolfgang Ziller
  • © Copyright 2005 by Frank Morgan
  • Journal: Proc. Amer. Math. Soc. 133 (2005), 2733-2735
  • MSC (2000): Primary 76U05, 49Q10, 85A30, 53C80
  • DOI: https://doi.org/10.1090/S0002-9939-05-08070-6
  • MathSciNet review: 2146221