Bonnesen's inequality

Short description: Relates the length, area and radius of the incircle and the circumcircle of a Jordan curve

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 [math]\displaystyle{ L }[/math] bounding a domain of area [math]\displaystyle{ A }[/math]. Let [math]\displaystyle{ r }[/math] and [math]\displaystyle{ R }[/math] denote the radii of the incircle and the circumcircle. Bonnesen proved the inequality [math]\displaystyle{ \pi^2 (R-r)^2 \leq L^2-4\pi A. }[/math]

The term [math]\displaystyle{ L^2-4\pi A }[/math] 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

• "Sur une amélioration de l'inégalité isopérimetrique du cercle et la démonstration d'une inégalité de Minkowski" (in fr), Comptes rendus hebdomadaires des séances de l'Académie des Sciences 172: 1087–1089, 1921, http://gallica.bnf.fr/ark:/12148/bpt6k3125x.f597 
• "1.3: The Bonnesen inequality and its analogues", Geometric Inequalities, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 285, Berlin: Springer-Verlag, 1988, pp. 3–4, doi:10.1007/978-3-662-07441-1, ISBN 3-540-13615-0