Riemann–Hilbert problem: Difference between revisions
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'''Bonnesen's inequality''' is an [[inequality (mathematics)|inequality]] relating the length, the area, the radius of the [[incircle]] and the radius of the [[circumcircle]] of a [[Jordan curve]]. It is a strengthening of the classical [[isoperimetry|isoperimetric inequality]]. | |||
More precisely, consider a planar simple closed curve of length <math>L</math> bounding a domain of area <math>F</math>. Let <math>r</math> and <math>R</math> denote the radii of the incircle and the circumcircle. Bonnesen proved the inequality | |||
:<math> L^2-4\pi F\geq \pi^2 (R-r)^2. \, </math> | |||
The term <math> \pi^2 (R-r)^2</math> in the right hand side is known as the <em>isoperimetric defect</em>. | |||
[[Loewner's torus inequality]] with isosystolic defect is a [[Systolic geometry|systolic]] analogue of Bonnesen's inequality. | |||
==References== | |||
* Bonnesen, T.: "Sur une amélioration de l'inégalité isopérimetrique du cercle et la démonstration d'une inégalité de Minkowski," ''C. R. Acad. Sci. Paris'' '''172''' (1921), 1087–1089. | |||
* Yu. D. Burago and V. A. Zalgaller, ''Geometric inequalities''. Translated from the Russian by A. B. Sosinskiĭ. Springer-Verlag, Berlin, 1988. ISBN 3-540-13615-0. | |||
[[Category:Elementary geometry]] | |||
[[Category:Geometric inequalities]] |
Revision as of 00:28, 22 January 2014
Bonnesen's inequality is an inequality relating the length, the area, the radius of the incircle and the radius of the circumcircle of a Jordan curve. It is a strengthening of the classical isoperimetric inequality.
More precisely, consider a planar simple closed curve of length bounding a domain of area . Let and denote the radii of the incircle and the circumcircle. Bonnesen proved the inequality
The term in the right hand side is known as the isoperimetric defect.
Loewner's torus inequality with isosystolic defect is a systolic analogue of Bonnesen's inequality.
References
- Bonnesen, T.: "Sur une amélioration de l'inégalité isopérimetrique du cercle et la démonstration d'une inégalité de Minkowski," C. R. Acad. Sci. Paris 172 (1921), 1087–1089.
- Yu. D. Burago and V. A. Zalgaller, Geometric inequalities. Translated from the Russian by A. B. Sosinskiĭ. Springer-Verlag, Berlin, 1988. ISBN 3-540-13615-0.