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]

References

^ ^a ^b 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
^ 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
^ Horowitz, Charles; Usadi Katz, Karin; Katz, Mikhail G. (2009), "Loewner's torus inequality with isosystolic defect", Journal of Geometric Analysis, 19 (4): 796–808, arXiv:0803.0690, doi:10.1007/s12220-009-9090-y, MR 2538936

This geometry-related article is a stub. You can help Wikipedia by expanding it.

[burago-zalgaller-1] 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

[bonnesen-2] 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

[hukk-3] Horowitz, Charles; Usadi Katz, Karin; Katz, Mikhail G. (2009), "Loewner's torus inequality with isosystolic defect", Journal of Geometric Analysis, 19 (4): 796–808, arXiv:0803.0690, doi:10.1007/s12220-009-9090-y, MR 2538936

[1]

[2]

[3]