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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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The ratio of the length of the unit circle to the area of the unit disc in Minkowski planes
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by Zokhrab Mustafaev PDF
Proc. Amer. Math. Soc. 133 (2005), 1231-1237 Request permission

Abstract:

In their paper “An Introduction to Finsler Geometry,” J. C. Alvarez and C. Duran asked if there are other Minkowski planes besides the Euclidean for which the ratio of the Minkowski length of the unit “circle” to the Holmes-Thompson area of the unit disc equals 2. In this paper we show that this ratio is greater than 2, and that the ratio 2 is achieved only for Minkowski planes that are affine equivalent to the Euclidean plane. In other words, the ratio is 2 only when the unit “circle” is an ellipse.
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Additional Information
  • Zokhrab Mustafaev
  • Affiliation: Department of Mathematics and Computer Science, 1212 Williams Hall, Ithaca College, Ithaca, New York 14850
  • Email: zmustafaev@ithaca.edu
  • Received by editor(s): October 28, 2003
  • Received by editor(s) in revised form: December 15, 2003
  • Published electronically: September 16, 2004
  • Communicated by: Jon G. Wolfson
  • © Copyright 2004 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 133 (2005), 1231-1237
  • MSC (2000): Primary 52A10, 52A40
  • DOI: https://doi.org/10.1090/S0002-9939-04-07662-2
  • MathSciNet review: 2117226