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.^[1]

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

The term $L^{2}-4\pi A$ in the right hand side is known as the isoperimetric defect.^[1]

Loewner's torus inequality with isosystolic defect is a systolic analogue of Bonnesen's inequality.^[3]

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