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The inequality between the volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i0528701.png" /> of a domain in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i0528702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i0528703.png" />, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i0528704.png" />-dimensional area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i0528705.png" /> of the hypersurface bounding the domain:
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<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/i/i052/i052870/i0528706.png" /></td> </tr></table>
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{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i0528707.png" /> is the volume of the unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i0528708.png" />-sphere. Equality holds only for a sphere. The classical isoperimetric inequality gives a solution of the [[Isoperimetric problem|isoperimetric problem]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i0528709.png" /> the classical isoperimetric inequality was known in Antiquity. A rigorous proof of the classical isoperimetric inequality for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287010.png" /> was given by F. Edler in 1882, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287011.png" /> by H.A. Schwarz in 1890, and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287012.png" /> by L.A. Lyusternik in 1935 and E. Schmidt in 1939 (see [[#References|[1]]], [[#References|[2]]], [[#References|[3]]]).
+
The inequality between the volume $  V $
 +
of a domain in a Euclidean space  $  \mathbf R  ^ {n} $,  
 +
$  n \geq  2 $,  
 +
and the  $  ( n - 1) $-dimensional area  $  F $
 +
of the hypersurface bounding the domain:
  
While in the two-dimensional case there are many proofs of the classical isoperimetric inequality (see [[#References|[4]]]), only two approaches are known for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287013.png" />. The first is the method of symmetrization proposed by J. Steiner. Using this method, Schmidt obtained analogues of the classical isoperimetric inequality (and the Brunn–Minkowski inequalities) for spherical and hyperbolic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287014.png" />-dimensional spaces (see [[#References|[5]]]). The second approach consists in reducing the classical isoperimetric inequality to a Brunn–Minkowski inequality (see [[Brunn–Minkowski theorem|Brunn–Minkowski theorem]]) and using the method of proportional division of volumes. In this approach there naturally arises the more general inequality
+
$$
 +
n  ^ {n} v _ {n} V ^ {n - 1 }  \leq  F ^ { n } ,
 +
$$
  
<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/i/i052/i052870/i05287015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
where  $  v _ {n} $
 +
is the volume of the unit  $  n $-sphere. Equality holds only for a sphere. The classical isoperimetric inequality gives a solution of the [[Isoperimetric problem|isoperimetric problem]]. For  $  n = 2, 3 $
 +
the classical isoperimetric inequality was known in Antiquity. A rigorous proof of the classical isoperimetric inequality for  $  n = 2 $
 +
was given by F. Edler in 1882, for  $  n = 3 $
 +
by H.A. Schwarz in 1890, and for all  $  n \geq  2 $
 +
by L.A. Lyusternik in 1935 and E. Schmidt in 1939 (see [[#References|[1]]], [[#References|[2]]], [[#References|[3]]]).
  
for volumes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287017.png" /> of two sets and the Minkowski area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287018.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287019.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287020.png" />. The inequality (*) can be interpreted as a classical isoperimetric inequality in Minkowski space; equality for a fixed Minkowski  "sphere"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287021.png" /> is not, generally speaking, attained for a unique body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287022.png" />; moreover, these bodies are different from a  "sphere"  (see [[#References|[6]]]).
+
While in the two-dimensional case there are many proofs of the classical isoperimetric inequality (see [[#References|[4]]]), only two approaches are known for $  n > 2 $.  
 +
The first is the method of symmetrization proposed by J. Steiner. Using this method, Schmidt obtained analogues of the classical isoperimetric inequality (and the Brunn–Minkowski inequalities) for spherical and hyperbolic  $  n $-dimensional spaces (see [[#References|[5]]]). The second approach consists in reducing the classical isoperimetric inequality to a Brunn–Minkowski inequality (see [[Brunn–Minkowski theorem|Brunn–Minkowski theorem]]) and using the method of proportional division of volumes. In this approach there naturally arises the more general inequality
 +
 
 +
$$ \tag{* }
 +
n  ^ {n} V ^ {n - 1 } ( A) V ( B)  \leq  F ^ { n } ( A, B)
 +
$$
 +
 
 +
for volumes  $  V ( A) $,
 +
$  V ( B) $
 +
of two sets and the Minkowski area $  F ( A, B) $
 +
of the set $  A $
 +
with respect to $  B $.  
 +
The inequality (*) can be interpreted as a classical isoperimetric inequality in Minkowski space; equality for a fixed Minkowski  "sphere"   $ B $
 +
is not, generally speaking, attained for a unique body $  A $;  
 +
moreover, these bodies are different from a  "sphere"  (see [[#References|[6]]]).
  
 
There are a number of generalizations of the classical isoperimetric inequality in which one does not consider domains with a piecewise-smooth boundary, but wider classes of sets, and the area of the boundary is considered in a generalized sense (Minkowski area, Lebesgue area, Caccioppoli–De Giorgi [[Perimeter|perimeter]] of a set, or the mass of a current, see [[#References|[7]]], [[#References|[8]]]). The classical isoperimetric inequality remains valid in all these cases, as well as for hypersurfaces with self-intersections and the corresponding oriented volume (see [[#References|[9]]]). These generalizations can be obtained from the classical isoperimetric inequality by limit transition for distinct variants of the concept of convergence.
 
There are a number of generalizations of the classical isoperimetric inequality in which one does not consider domains with a piecewise-smooth boundary, but wider classes of sets, and the area of the boundary is considered in a generalized sense (Minkowski area, Lebesgue area, Caccioppoli–De Giorgi [[Perimeter|perimeter]] of a set, or the mass of a current, see [[#References|[7]]], [[#References|[8]]]). The classical isoperimetric inequality remains valid in all these cases, as well as for hypersurfaces with self-intersections and the corresponding oriented volume (see [[#References|[9]]]). These generalizations can be obtained from the classical isoperimetric inequality by limit transition for distinct variants of the concept of convergence.
  
For the isoperimetric difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287023.png" />, and the isoperimetric ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287024.png" />, estimates are known which strengthen the classical isoperimetric inequality (see [[#References|[2]]]). Some of these estimates are obtained for sets of special shape, in the first place for convex sets (cf. [[Convex set|Convex set]]) and polyhedra (see [[#References|[10]]]). An example of this is the [[Bonnesen inequality|Bonnesen inequality]] for plane figures:
+
For the isoperimetric difference $  F ^ { n } - n  ^ {n} v _ {n} V ^ {n - 1 } $,  
 +
and the isoperimetric ratio $  F ^ { n } V ^ {1 - n } $,  
 +
estimates are known which strengthen the classical isoperimetric inequality (see [[#References|[2]]]). Some of these estimates are obtained for sets of special shape, in the first place for convex sets (cf. [[Convex set|Convex set]]) and polyhedra (see [[#References|[10]]]). An example of this is the [[Bonnesen inequality|Bonnesen inequality]] for plane figures:
  
<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/i/i052/i052870/i05287025.png" /></td> </tr></table>
+
$$
 +
F ^ { 2 } - 4 \pi V  \geq  ( F - 4 \pi r)  ^ {2} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287026.png" /> is the radius of the largest inscribed circle, and its generalization (see [[#References|[11]]]) for convex bodies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287027.png" />:
+
where $  r $
 +
is the radius of the largest inscribed circle, and its generalization (see [[#References|[11]]]) for convex bodies in $  \mathbf R  ^ {n} $:
  
<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/i/i052/i052870/i05287028.png" /></td> </tr></table>
+
$$
 +
F ^ { n/( n - 1) } ( A, B) - n ^ {n/( n - 1) } V ( A)
 +
V ^ {1/( n - 1) } ( B) \geq
 +
$$
  
<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/i/i052/i052870/i05287029.png" /></td> </tr></table>
+
$$
 +
\geq  \
 +
[ F ( A, B) -
 +
n ^ {n/( n - 1) } qV ( B) ^ {1/( n - 1) } ]  ^ {n} .
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287030.png" />. The relative isoperimetric difference of two convex bodies,
+
Here $  q = \max \{  \lambda  : {\lambda B  \textrm{ can  be  imbedded  in  }  A } \} $.  
 +
The relative isoperimetric difference of two convex bodies,
  
<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/i/i052/i052870/i05287031.png" /></td> </tr></table>
+
$$
 +
F ^ { n } ( A, B) - n  ^ {n} V ^ {n - 1 } ( A) V ( B) ,
 +
$$
  
 
can serve as a measure of their non-homotheticity (see [[#References|[12]]]). It is used, for example, in proving stability theorems in the [[Minkowski problem|Minkowski problem]] (see [[#References|[13]]]). For generalizations of the classical isoperimetric inequality to spaces of variable curvature and related inequalities, see [[Isoperimetric inequality|Isoperimetric inequality]].
 
can serve as a measure of their non-homotheticity (see [[#References|[12]]]). It is used, for example, in proving stability theorems in the [[Minkowski problem|Minkowski problem]] (see [[#References|[13]]]). For generalizations of the classical isoperimetric inequality to spaces of variable curvature and related inequalities, see [[Isoperimetric inequality|Isoperimetric inequality]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.A. Kryzhanovskii,  "Isoperimeters" , Moscow  (1959)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Hadwiger,  "Vorlesungen über Inhalt, Oberfläche und Isoperimetrie" , Springer  (1957)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.A. Lyusternik,  "Application of the Brunn–Minkowski inequality to extremal problems"  ''Uspekhi Mat. Nauk'' , '''2'''  (1936)  pp. 47–54  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H. Reichardt,  "Einführung in die Differentialgeometrie" , Springer  (1960)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E. Schmidt,  "Die Brunn–Minkowskische Ungleichung und ihr Spiegelbild sowie die isoperimetrische Eigenschaft der Kugel in der euklidischen und nichteuklidischen Geometrie I"  ''Math. Nachr.'' , '''1'''  (1948)  pp. 81–157</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  H. Busemann,  "The isoperimetric problem for Minkowski area"  ''Amer. J. Math.'' , '''71'''  (1949)  pp. 743–762</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E. De Giorgi,  "Sulla proprietà isoperimetrica dell'ipersfera, nella classe degli insience aventi frontiera orientata di misura finita"  ''Atti Acad. Naz. Lincei Mem. Cl. Sci. Fis., Mat. e Natur.'' , '''8''' :  5, 2  (1958)  pp. 33–44</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  H. Federer,  W.H. Fleming,  "Normal and integer currents"  ''Ann. of Math. (2)'' , '''72'''  (1960)  pp. 458–520</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  T. Radó,  "The isoperimetric inequality and the Lebesgue definition of surface area"  ''Trans. Amer. Math. Soc.'' , '''61''' :  3  (1947)  pp. 530–555</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  L. Fejes Toth,  "Lagerungen in der Ebene, auf der Kugel und im Raum" , Springer  (1972)</TD></TR><TR><TD valign="top">[11]</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><TR><TD valign="top">[12]</TD> <TD valign="top">  V.I. Diskant,  "Bounds for the discrepancy between convex bodies in terms of the isoperimetric difference"  ''Siberian Math. J.'' , '''13''' :  4  (1973)  pp. 529–532  ''Sibirsk. Mat. Zh.'' , '''13''' :  4  (1972)  pp. 767–772</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  Yu.A. Volkov,  "Stability of the solution to Minkowski's problem"  ''Vestnik Leningrad. Univ. Ser. Mat. Astron.'' , '''18'''  (1963)  pp. 33–43  (In Russian)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.A. Kryzhanovskii,  "Isoperimeters" , Moscow  (1959)  (In Russian) {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Hadwiger,  "Vorlesungen über Inhalt, Oberfläche und Isoperimetrie" , Springer  (1957) {{MR|0102775}} {{ZBL|0078.35703}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.A. Lyusternik,  "Application of the Brunn–Minkowski inequality to extremal problems"  ''Uspekhi Mat. Nauk'' , '''2'''  (1936)  pp. 47–54  (In Russian) {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H. Reichardt,  "Einführung in die Differentialgeometrie" , Springer  (1960) {{MR|0116267}} {{ZBL|0091.34001}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E. Schmidt,  "Die Brunn–Minkowskische Ungleichung und ihr Spiegelbild sowie die isoperimetrische Eigenschaft der Kugel in der euklidischen und nichteuklidischen Geometrie I"  ''Math. Nachr.'' , '''1'''  (1948)  pp. 81–157 {{MR|0028600}} {{ZBL|0030.07602}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  H. Busemann,  "The isoperimetric problem for Minkowski area"  ''Amer. J. Math.'' , '''71'''  (1949)  pp. 743–762 {{MR|0031762}} {{ZBL|0038.10301}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E. De Giorgi,  "Sulla proprietà isoperimetrica dell'ipersfera, nella classe degli insience aventi frontiera orientata di misura finita"  ''Atti Acad. Naz. Lincei Mem. Cl. Sci. Fis., Mat. e Natur.'' , '''8''' :  5, 2  (1958)  pp. 33–44 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  H. Federer,  W.H. Fleming,  "Normal and integer currents"  ''Ann. of Math. (2)'' , '''72'''  (1960)  pp. 458–520 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  T. Radó,  "The isoperimetric inequality and the Lebesgue definition of surface area"  ''Trans. Amer. Math. Soc.'' , '''61''' :  3  (1947)  pp. 530–555 {{MR|0021966}} {{ZBL|0035.32601}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  L. Fejes Toth,  "Lagerungen in der Ebene, auf der Kugel und im Raum" , Springer  (1972) {{MR|}} {{ZBL|0229.52009}} </TD></TR><TR><TD valign="top">[11]</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 {{MR|338925}} {{ZBL|}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  V.I. Diskant,  "Bounds for the discrepancy between convex bodies in terms of the isoperimetric difference"  ''Siberian Math. J.'' , '''13''' :  4  (1973)  pp. 529–532  ''Sibirsk. Mat. Zh.'' , '''13''' :  4  (1972)  pp. 767–772 {{MR|}} {{ZBL|0266.52008}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  Yu.A. Volkov,  "Stability of the solution to Minkowski's problem"  ''Vestnik Leningrad. Univ. Ser. Mat. Astron.'' , '''18'''  (1963)  pp. 33–43  (In Russian) {{MR|}} {{ZBL|}} </TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
The Minkowski area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287032.png" /> of a convex set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287033.png" /> with respect to a convex set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287034.png" /> is defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287035.png" /> be the support function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287036.png" />, i.e. for each vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287038.png" /> defines a supporting plane of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287039.png" /> such that the open half-space into which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287040.png" /> points contains no points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287041.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287042.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287043.png" /> (and equality holds for at least one point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287044.png" />). The Minkowski area of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287045.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287046.png" /> is now defined by
+
The Minkowski area $  F ( A , B ) $
 +
of a convex set $  A $
 +
with respect to a convex set $  B $
 +
is defined as follows. Let $  H _ {B} ( u) $
 +
be the support function of $  B $,  
 +
i.e. for each vector $  u \in \mathbf R  ^ {n} $,
 +
$  \sum_{i=1}^ {n} u _ {i} x _ {i} = H _ {B} ( u) $
 +
defines a supporting plane of $  B $
 +
such that the open half-space into which $  u $
 +
points contains no points of $  B $,  
 +
so that $  \sum u _ {i} x _ {i} \leq  H _ {B} ( u) $
 +
for all $  x \in B $ (and equality holds for at least one point of $  B $).  
 +
The Minkowski area of $  A $
 +
with respect to $  B $
 +
is now defined by
  
<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/i/i052/i052870/i05287047.png" /></td> </tr></table>
+
$$
 +
F ( A , B )  = \int\limits _ { S } H _ {B} ( u)  d S
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287048.png" /> is the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287049.png" />. It is also equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287050.png" /> times the mixed volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287051.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287052.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287053.png" />'s). Here the mixed volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287054.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287055.png" /> convex sets is defined as the coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287056.png" /> in the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052870/i05287057.png" />.
+
where $  S $
 +
is the boundary of $  A $.  
 +
It is also equal to $  n $
 +
times the mixed volume $  V ( A , B \dots B ) $ ($  ( n - 1) $
 +
$  B  $'
 +
s). Here the mixed volume $  V ( A _ {1} \dots A _ {n} ) $
 +
of $  n $
 +
convex sets is defined as the coefficient of $  \lambda _ {1} \dots \lambda _ {n} $
 +
in the polynomial $  V ( \lambda _ {1} A _ {1} + \dots + \lambda _ {n} A _ {n} ) $.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Bonnesen,  W. Fenchel,  "Theorie der konvexen Körper" , Chelsea, reprint  (1948)  pp. Sects. 15, 29, 31, 38</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Bonnesen,  W. Fenchel,  "Theorie der konvexen Körper" , Chelsea, reprint  (1948)  pp. Sects. 15, 29, 31, 38 {{MR|0344997}} {{MR|0372748}} {{MR|1512278}} {{ZBL|0277.52001}} {{ZBL|0906.52001}} {{ZBL|0008.07708}}  {{ZBL|60.0673.01}}  {{ZBL|51.0373.01}} </TD></TR></table>

Latest revision as of 08:14, 13 January 2024


The inequality between the volume $ V $ of a domain in a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, and the $ ( n - 1) $-dimensional area $ F $ of the hypersurface bounding the domain:

$$ n ^ {n} v _ {n} V ^ {n - 1 } \leq F ^ { n } , $$

where $ v _ {n} $ is the volume of the unit $ n $-sphere. Equality holds only for a sphere. The classical isoperimetric inequality gives a solution of the isoperimetric problem. For $ n = 2, 3 $ the classical isoperimetric inequality was known in Antiquity. A rigorous proof of the classical isoperimetric inequality for $ n = 2 $ was given by F. Edler in 1882, for $ n = 3 $ by H.A. Schwarz in 1890, and for all $ n \geq 2 $ by L.A. Lyusternik in 1935 and E. Schmidt in 1939 (see [1], [2], [3]).

While in the two-dimensional case there are many proofs of the classical isoperimetric inequality (see [4]), only two approaches are known for $ n > 2 $. The first is the method of symmetrization proposed by J. Steiner. Using this method, Schmidt obtained analogues of the classical isoperimetric inequality (and the Brunn–Minkowski inequalities) for spherical and hyperbolic $ n $-dimensional spaces (see [5]). The second approach consists in reducing the classical isoperimetric inequality to a Brunn–Minkowski inequality (see Brunn–Minkowski theorem) and using the method of proportional division of volumes. In this approach there naturally arises the more general inequality

$$ \tag{* } n ^ {n} V ^ {n - 1 } ( A) V ( B) \leq F ^ { n } ( A, B) $$

for volumes $ V ( A) $, $ V ( B) $ of two sets and the Minkowski area $ F ( A, B) $ of the set $ A $ with respect to $ B $. The inequality (*) can be interpreted as a classical isoperimetric inequality in Minkowski space; equality for a fixed Minkowski "sphere" $ B $ is not, generally speaking, attained for a unique body $ A $; moreover, these bodies are different from a "sphere" (see [6]).

There are a number of generalizations of the classical isoperimetric inequality in which one does not consider domains with a piecewise-smooth boundary, but wider classes of sets, and the area of the boundary is considered in a generalized sense (Minkowski area, Lebesgue area, Caccioppoli–De Giorgi perimeter of a set, or the mass of a current, see [7], [8]). The classical isoperimetric inequality remains valid in all these cases, as well as for hypersurfaces with self-intersections and the corresponding oriented volume (see [9]). These generalizations can be obtained from the classical isoperimetric inequality by limit transition for distinct variants of the concept of convergence.

For the isoperimetric difference $ F ^ { n } - n ^ {n} v _ {n} V ^ {n - 1 } $, and the isoperimetric ratio $ F ^ { n } V ^ {1 - n } $, estimates are known which strengthen the classical isoperimetric inequality (see [2]). Some of these estimates are obtained for sets of special shape, in the first place for convex sets (cf. Convex set) and polyhedra (see [10]). An example of this is the Bonnesen inequality for plane figures:

$$ F ^ { 2 } - 4 \pi V \geq ( F - 4 \pi r) ^ {2} , $$

where $ r $ is the radius of the largest inscribed circle, and its generalization (see [11]) for convex bodies in $ \mathbf R ^ {n} $:

$$ F ^ { n/( n - 1) } ( A, B) - n ^ {n/( n - 1) } V ( A) V ^ {1/( n - 1) } ( B) \geq $$

$$ \geq \ [ F ( A, B) - n ^ {n/( n - 1) } qV ( B) ^ {1/( n - 1) } ] ^ {n} . $$

Here $ q = \max \{ \lambda : {\lambda B \textrm{ can be imbedded in } A } \} $. The relative isoperimetric difference of two convex bodies,

$$ F ^ { n } ( A, B) - n ^ {n} V ^ {n - 1 } ( A) V ( B) , $$

can serve as a measure of their non-homotheticity (see [12]). It is used, for example, in proving stability theorems in the Minkowski problem (see [13]). For generalizations of the classical isoperimetric inequality to spaces of variable curvature and related inequalities, see Isoperimetric inequality.

References

[1] D.A. Kryzhanovskii, "Isoperimeters" , Moscow (1959) (In Russian)
[2] H. Hadwiger, "Vorlesungen über Inhalt, Oberfläche und Isoperimetrie" , Springer (1957) MR0102775 Zbl 0078.35703
[3] L.A. Lyusternik, "Application of the Brunn–Minkowski inequality to extremal problems" Uspekhi Mat. Nauk , 2 (1936) pp. 47–54 (In Russian)
[4] H. Reichardt, "Einführung in die Differentialgeometrie" , Springer (1960) MR0116267 Zbl 0091.34001
[5] E. Schmidt, "Die Brunn–Minkowskische Ungleichung und ihr Spiegelbild sowie die isoperimetrische Eigenschaft der Kugel in der euklidischen und nichteuklidischen Geometrie I" Math. Nachr. , 1 (1948) pp. 81–157 MR0028600 Zbl 0030.07602
[6] H. Busemann, "The isoperimetric problem for Minkowski area" Amer. J. Math. , 71 (1949) pp. 743–762 MR0031762 Zbl 0038.10301
[7] E. De Giorgi, "Sulla proprietà isoperimetrica dell'ipersfera, nella classe degli insience aventi frontiera orientata di misura finita" Atti Acad. Naz. Lincei Mem. Cl. Sci. Fis., Mat. e Natur. , 8 : 5, 2 (1958) pp. 33–44
[8] H. Federer, W.H. Fleming, "Normal and integer currents" Ann. of Math. (2) , 72 (1960) pp. 458–520
[9] T. Radó, "The isoperimetric inequality and the Lebesgue definition of surface area" Trans. Amer. Math. Soc. , 61 : 3 (1947) pp. 530–555 MR0021966 Zbl 0035.32601
[10] L. Fejes Toth, "Lagerungen in der Ebene, auf der Kugel und im Raum" , Springer (1972) Zbl 0229.52009
[11] 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 MR338925
[12] V.I. Diskant, "Bounds for the discrepancy between convex bodies in terms of the isoperimetric difference" Siberian Math. J. , 13 : 4 (1973) pp. 529–532 Sibirsk. Mat. Zh. , 13 : 4 (1972) pp. 767–772 Zbl 0266.52008
[13] Yu.A. Volkov, "Stability of the solution to Minkowski's problem" Vestnik Leningrad. Univ. Ser. Mat. Astron. , 18 (1963) pp. 33–43 (In Russian)

Comments

The Minkowski area $ F ( A , B ) $ of a convex set $ A $ with respect to a convex set $ B $ is defined as follows. Let $ H _ {B} ( u) $ be the support function of $ B $, i.e. for each vector $ u \in \mathbf R ^ {n} $, $ \sum_{i=1}^ {n} u _ {i} x _ {i} = H _ {B} ( u) $ defines a supporting plane of $ B $ such that the open half-space into which $ u $ points contains no points of $ B $, so that $ \sum u _ {i} x _ {i} \leq H _ {B} ( u) $ for all $ x \in B $ (and equality holds for at least one point of $ B $). The Minkowski area of $ A $ with respect to $ B $ is now defined by

$$ F ( A , B ) = \int\limits _ { S } H _ {B} ( u) d S $$

where $ S $ is the boundary of $ A $. It is also equal to $ n $ times the mixed volume $ V ( A , B \dots B ) $ ($ ( n - 1) $ $ B $' s). Here the mixed volume $ V ( A _ {1} \dots A _ {n} ) $ of $ n $ convex sets is defined as the coefficient of $ \lambda _ {1} \dots \lambda _ {n} $ in the polynomial $ V ( \lambda _ {1} A _ {1} + \dots + \lambda _ {n} A _ {n} ) $.

References

[a1] T. Bonnesen, W. Fenchel, "Theorie der konvexen Körper" , Chelsea, reprint (1948) pp. Sects. 15, 29, 31, 38 MR0344997 MR0372748 MR1512278 Zbl 0277.52001 Zbl 0906.52001 Zbl 0008.07708 Zbl 60.0673.01 Zbl 51.0373.01
How to Cite This Entry:
Isoperimetric inequality, classical. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isoperimetric_inequality,_classical&oldid=15338
This article was adapted from an original article by Yu.D. Burago (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article