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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016890/b0168901.png" /> be a [[Convex domain|convex domain]] in the plane, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016890/b0168902.png" /> be the radius of the largest circle which can be inserted in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016890/b0168903.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016890/b0168904.png" /> be the radius of the smallest circle containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016890/b0168905.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016890/b0168906.png" /> be the perimeter and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016890/b0168907.png" /> be the area of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016890/b0168908.png" />. The Bonnesen inequality [[#References|[1]]]
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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]]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016890/b0168909.png" /></td> </tr></table>
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$$\Delta=L^2-4\pi F\geq\pi^2(R-r)^2$$
  
is then valid. The equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016890/b01689010.png" /> is attained only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016890/b01689011.png" />, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016890/b01689012.png" /> is a disc. For generalizations of the Bonnesen inequality see [[#References|[2]]].
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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=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