Riemann–Hilbert problem

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

$L^{2}-4\pi F\geq \pi ^{2}(R-r)^{2}.\,$

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