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Proceedings of the American Mathematical Society

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

 

 

An application of the maximum principle to the geometry of plane curves


Author: Harold H. Johnson
Journal: Proc. Amer. Math. Soc. 44 (1974), 432-435
MSC: Primary 52A40; Secondary 49B10
MathSciNet review: 0348631
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Abstract: The maximum principle of control theory is used to find necessary and sufficient conditions for a plane curve, which has bounded piecewise continuous curvature and prescribed initial and terminal points and directions, to have minimal length. This result is used to prove that such a closed curve having length $ L$ and curvature $ k$ satisfying $ \vert k\vert \leqq K$ can be contained in a circle of radius $ R$, where $ R \leqq L/4 - (\pi - 2)/2K$.


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

  • [1] V. G. \cyr{B}oltyanskiĭ, Matematicheskie metody optimalnogo upravleniya, Second revised and supplemented edition, Izdat. “Nauka”, Moscow, 1969 (Russian). Physico-Mathematical Library for the Engineer. MR 0353082
  • [2] J. C. C. Nitsche, The smallest sphere containing a rectifiable curve, Amer. Math. Monthly 78 (1971), 881–882. MR 0291387

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0348631-6
Keywords: Plane curves, maximum principle, geometric inequalities
Article copyright: © Copyright 1974 American Mathematical Society