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 ${\displaystyle L}$ bounding a domain of area ${\displaystyle A}$. Let ${\displaystyle r}$ and ${\displaystyle R}$ denote the radii of the incircle and the circumcircle. Bonnesen proved the inequality

${\displaystyle \pi ^{2}(R-r)^{2}\leq L^{2}-4\pi A.\,}$

The term ${\displaystyle \pi ^{2}(R-r)^{2}}$ 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.

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.