|
The ratio of the length of the unit circle to the area of the unit disc in Minkowski planes
Author(s):
Zokhrab
Mustafaev
Journal:
Proc. Amer. Math. Soc.
133
(2005),
1231-1237.
MSC (2000):
Primary 52A10, 52A40
Posted:
September 16, 2004
MathSciNet review:
2117226
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
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.
References:
-
- 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)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (2000):
52A10, 52A40
Retrieve articles in all Journals with
MSC (2000):
52A10, 52A40
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:
10.1090/S0002-9939-04-07662-2
PII:
S 0002-9939(04)07662-2
Received by editor(s):
October 28, 2003
Received by editor(s) in revised form:
December 15, 2003
Posted:
September 16, 2004
Communicated by:
Jon G. Wolfson
Copyright of article:
Copyright
2004,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
