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.

References

• Bonnesen, T. (1921), "Sur une amélioration de l'inégalité isopérimetrique du cercle et la démonstration d'une inégalité de Minkowski", Comptes rendus hebdomadaires des séances de l'Académie des Sciences (in French), 172: 1087–1089, JFM 48.0839.01
• Burago, Yu. D.; Zalgaller, V. A. (1988), "1.3: The Bonnesen inequality and its analogues", Geometric Inequalities, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, translated by Sosinskiĭ, A. B., Berlin: Springer-Verlag, pp. 3–4, doi:10.1007/978-3-662-07441-1, ISBN 3-540-13615-0, MR 0936419, Zbl 0633.53002

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This page was last edited on 12 May 2024, at 20:51