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

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

The term ${\displaystyle L^{2}-4\pi A}$ 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.