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]].
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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]]

Latest revision as of 10:14, 18 November 2014

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