Riemann–Hilbert problem: Difference between revisions

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 $L$ bounding a domain of area $F$ . Let $r$ and $R$ denote the radii of the incircle and the circumcircle. Bonnesen proved the inequality

L^{2} - 4 π F \geq π^{2} (R - r)^{2} .

The term $π^{2} (R - r)^{2}$ 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.

@@ Line 1: / Line 1: @@
-Friends call her Felicidad and her husband doesn't like it at all. Camping is some thing that I've done for many years. Interviewing is what she does in her day job but quickly her spouse and her will begin their own company. Her family lives in Idaho.<br><br>Here is my web blog :: extended car warranty - [http://Chaddibaddi.com/blogs/post/24854 Click at Chaddibaddi.com] -
+'''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]]

Riemann–Hilbert problem: Difference between revisions

Revision as of 00:28, 22 January 2014

References

Navigation menu

Search