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

Isoperimetry

The isoperimetric inequality is a geometric inequality involving the square of the circumference of a closed curve in the plane and the area of a plane region it encloses, as well as its various generalizations. Isoperimetric literally means "having the same perimeter"...

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More precisely, consider a planar simple closed curve of length bounding a domain of area . Let and denote the radii of the incircle and the circumcircle. Bonnesen proved the inequality


The term in the right hand side is known as the isoperimetric defect.

Loewner's torus inequality

Loewner's torus inequality

In differential geometry, Loewner's torus inequality is an inequality due to Charles Loewner. It relates the systole and the area of an arbitrary Riemannian metric on the 2-torus.-Statement:...

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