Isoperimetric inequality, classical

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