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 $$ \pi^2 (R-r)^2 \leq L^2-4\pi A. $$

The term $$ L^2-4\pi A$$ in the right hand side is known as the isoperimetric defect.

Loewner's torus inequality with isosystolic defect is a systolic analogue of Bonnesen's inequality.