The ratio of the length of the unit circle to the area of the unit disc in Minkowski planes

Author:
Zokhrab Mustafaev

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

Published electronically:
September 16, 2004

MathSciNet review:
2117226

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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.

**1.**Alvarez, J. and Duran, C., ``An introduction to Finsler Geometry", Notas de la Escuela Venezolana de Mathematicas, 1998.**2.**Eggleston, H. G., ``Convexity", Cambridge Univ. Tracts in Math. and Math. Phys. 47, Cambridge (1958). MR**0124813 (23:A2123)****3.**Holmes, R. D. and Thompson, A. C.,*-dimensional area and content in Minkowski spaces*, Pacific J. Math. 85 (1979), 77-110. MR**0571628 (81k:52023)****4.**Reisner, S.,*Zonoids with minimal volume product,*Math. Z. 192 (1986), 339-346. MR**0845207 (87g:52022)****5.**Schneider, R., ``Convex Bodies: The Brunn-Minkowski Theory", Encyclopedia of Math. and Its Appl. 44, Cambridge Univ. Press, Cambridge (1993). MR**1216521 (94d:52007)****6.**Thompson, A. C., ``Minkowski Geometry", Encyclopedia of Math. and Its Appl., Vol.63, Cambridge Univ. Press, Cambridge, 1996. MR**1406315 (97f:52001)**

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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

DOI:
https://doi.org/10.1090/S0002-9939-04-07662-2

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

Article copyright:
© Copyright 2004
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.