Bonnesen inequality

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

$$\Delta=L^2-4\pi F\geq\pi^2(R-r)^2$$

is then valid. The equality $\Delta=0$ is attained only if $R=r$, i.e. if $K$ 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