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

The term π 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.