Bonnesen inequality

One of the more precise forms of the isoperimetric inequality for convex domains in the plane. Let be a convex domain in the plane, let be the radius of the largest circle which can be inserted in , let be the radius of the smallest circle containing , let be the perimeter and let be the area of . The Bonnesen inequality [1]

is then valid. The equality is attained only if , i.e. if is a disc. For generalizations of the Bonnesen inequality see [2].

References

[1]	T. Bonnesen, "Ueber eine Verschärferung der isoperimetische Ungleichheit des Kreises in der Ebene und auf die Kugeloberfläche nebst einer Anwendung auf eine Minkowskische Ungleichheit für konvexe Körper" Math. Ann. , 84 (1921) pp. 216–227
[2]	V.I. Diskant, "A generalization of Bonnesen's inequalities" Soviet Math. Dokl. , 14 : 6 (1973) pp. 1728–1731 Dokl. Akad. Nauk SSSR , 213 : 3 (1973) pp. 519–521