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.
[1] More precisely, consider a planar simple closed curve of length
denote the radii of the incircle and the circumcircle.
in the right hand side is known as the isoperimetric defect.
[1] Loewner's torus inequality with isosystolic defect is a systolic analogue of Bonnesen's inequality.