Difference between revisions of "Bonnesen inequality"
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| − | One of the more precise forms of the [[Isoperimetric inequality|isoperimetric inequality]] for convex domains in the plane. Let | + | {{TEX|done}} |
| + | One of the more precise forms of the [[Isoperimetric inequality|isoperimetric inequality]] for convex domains in the plane. Let $K$ be a [[Convex domain|convex domain]] in the plane, let $r$ be the radius of the largest circle which can be inserted in $K$, let $R$ be the radius of the smallest circle containing $K$, let $L$ be the perimeter and let $F$ be the area of $K$. The Bonnesen inequality [[#References|[1]]] | ||
| − | + | $$\Delta=L^2-4\pi F\geq\pi^2(R-r)^2$$ | |
| − | is then valid. The equality | + | is then valid. The equality $\Delta=0$ is attained only if $R=r$, i.e. if $K$ is a disc. For generalizations of the Bonnesen inequality see [[#References|[2]]]. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> 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</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> 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</TD></TR></table> | ||
Latest revision as of 18:16, 16 April 2014
One of the more precise forms of the isoperimetric inequality for convex domains in the plane. Let $K$ be a convex domain in the plane, let $r$ be the radius of the largest circle which can be inserted in $K$, let $R$ be the radius of the smallest circle containing $K$, let $L$ be the perimeter and let $F$ be the area of $K$. The Bonnesen inequality [1]
$$\Delta=L^2-4\pi F\geq\pi^2(R-r)^2$$
is then valid. The equality $\Delta=0$ is attained only if $R=r$, i.e. if $K$ 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=31788
Bonnesen inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bonnesen_inequality&oldid=31788
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article