Bonnesen's inequality

Template:Short description 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.Template:R

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 inequalityTemplate:R $${\pi }^2(R-r{)}^2\le L^2-4\pi A.$$

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

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

References

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