Bonnesen inequality
From Encyclopedia of Mathematics
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One of the more precise forms of the isoperimetric inequality for convex domains in the plane. Let 
be a convex domain in the plane, let 
be the radius of the largest circle which can be inserted in 
, let 
be the radius of the smallest circle containing 
, let 
be the perimeter and let 
be the area of 
. The Bonnesen inequality [1]
 |
is then valid. The equality 
is attained only if 
, i.e. if 
is a disc. For generalizations of the Bonnesen inequality see [2].
References
| [1] | T. Bonnesen, "Ueber eine Verschärferung der isoperimetische Ungleichheit des Kreises in der Ebene und auf die Kugeloberfläche nebst einer Anwendung auf eine Minkowskische Ungleichheit für konvexe Körper" Math. Ann. , 84 (1921) pp. 216–227 |
| [2] | V.I. Diskant, "A generalization of Bonnesen's inequalities" Soviet Math. Dokl. , 14 : 6 (1973) pp. 1728–1731 Dokl. Akad. Nauk SSSR , 213 : 3 (1973) pp. 519–521 |
How to Cite This Entry:
Bonnesen inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bonnesen_inequality&oldid=16744
Bonnesen inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bonnesen_inequality&oldid=16744
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article