Bonnesen's inequality

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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 A. Let r and R denote the radii of the incircle and the circumcircle. Bonnesen proved the inequality

 \pi^2 (R-r)^2 \leq L^2-4\pi A. \,

The term  \pi^2 (R-r)^2 in the left hand side is known as the isoperimetric defect.

Loewner's torus inequality with isosystolic defect is a systolic analogue of Bonnesen's inequality.


  • 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.