THE BONNESEN-TYPE INEQUALITIES IN A PLANE OF CONSTANT CURVATURE

- Journal title : Journal of the Korean Mathematical Society
- Volume 44, Issue 6, 2007, pp.1363-1372
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2007.44.6.1363

Title & Authors

THE BONNESEN-TYPE INEQUALITIES IN A PLANE OF CONSTANT CURVATURE

Zhou, Jiazu; Chen, Fangwei;

Zhou, Jiazu; Chen, Fangwei;

Abstract

We investigate the containment measure of one domain to contain in another domain in a plane of constant curvature. We obtain some Bonnesen-type inequalities involving the area, length, radius of the inscribed and the circumscribed disc of a domain D in .

Keywords

isoperimetric inequality;Bonessen inequality;kinematic measure;containment measure;hyperbolic plane;projective plane;geodesic disc;

Language

English

Cited by

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References

1.

Y. D. Burago and V. A. Zalgaller, Geometric inequalities, Translated from the Rus-sian by A. B. Sosinskii. Grundlehren der Mathematischen Wissenschaften, 285. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1988

2.

E. Grinberg, D. Ren, and J. Zhou, The symmetric isoperimetric deficit and the contain- ment problem in a plane of constant curvature, preprint

3.

E. Grinberg, G. Zhang, J. Zhou, and S. Li, Integral geometry and convexity, Proceedings of the 1st International Conference on Integral Geometry and Convexity Related Topics held at Wuhan University of Science and Technology, Wuhan, October 18-23, 2004. Edited by Eric L. Grinberg, Shougui Li, Gaoyong Zhang and Jiazu Zhou.World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006

4.

D. Ren, Topics in integral geometry, Translated from the Chinese and revised by the author. With forewords by Shiing Shen Chern and Chuan-Chih Hsiung. Series in Pure Mathematics, 19. World Scientific Publishing Co., Inc., River Edge, NJ, 1994

5.

L. A. Santalo, Integral geometry and geometric probability, With a foreword by Mark Kac. Encyclopedia of Mathematics and its Applications, Vol. 1. Addison-Wesley Pub- lishing Co., Reading, Mass.-London-Amsterdam, 1976

6.

R. Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathemat- ics and its Applications, 44. Cambridge University Press, Cambridge, 1993

7.

G. Zhang, A suffcient condition for one convex body containing another, Chinese Ann. Math. Ser. B 9 (1988), no. 4, 447-451

8.

G. Zhang and J. Zhou, Containment measures in integral geometry, Integral geometry and convexity, 153-168, World Sci. Publ., Hackensack, NJ, 2006

9.

J. Zhou, On Bonnesen-type inequalities, Acta Math. Sin. 50, No. 6, 2007

10.

J. Zhou, When can one domain enclose another in R3?, J. Austral. Math. Soc. Ser. A 59 (1995), no. 2, 266-272

11.

J. Zhou, The suffcient condition for a convex body to enclose another in R4, Proc. Amer. Math. Soc. 121 (1994), no. 3, 907-913

12.

J. Zhou, Suffcient conditions for one domain to contain another in a space of constant curvature, Proc. Amer. Math. Soc. 126 (1998), no. 9, 2797-2803

13.

J.Zhou, Total square mean curvature of hypersurfaces, preprint submitted