Bonnesen's inequality

Relates the length, area and radius of the incircle and the circumcircle of a Jordan curve

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 {\displaystyle L} bounding a domain of area A {\displaystyle A} . Let r {\displaystyle r} and R {\displaystyle R} denote the radii of the incircle and the circumcircle. Bonnesen proved the inequality[2] π 2 ( R − r ) 2 ≤ L 2 − 4 π A . {\displaystyle \pi ^{2}(R-r)^{2}\leq L^{2}-4\pi A.}

The term L 2 − 4 π A {\displaystyle 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

  1. ^ 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
  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
  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


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