View source for Isoperimetric inequality, classical

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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|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]]]).

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.

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:

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

where  $  r $
is the radius of the largest inscribed circle, and its generalization (see [[#References|[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 [[#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====
<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====
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====
<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>