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 bounding a domain of area. Let and denote the radii of the incircle and the circumcircle. Bonnesen proved the inequality
The term 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.
