# Bonnesen's inequality

**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 left 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.

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