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 {\displaystyle L} bounding a domain of area A {\displaystyle A} . Let r {\displaystyle r} and R {\displaystyle R} denote the radii of the incircle and the circumcircle. Bonnesen proved the inequality
π 2 ( R − r ) 2 ≤ L 2 − 4 π A . {\displaystyle \pi ^{2}(R-r)^{2}\leq L^{2}-4\pi A.}
The term L 2 − 4 π A {\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.
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