View source for Isoperimetric inequality

''(in geometry and physics)''

A general term referring to the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i0528601.png" /> between the area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i0528602.png" /> and perimeter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i0528603.png" /> of a plane domain, to its various generalizations and to other inequalities between geometrical characteristics of figures, sets and manifolds. Also belonging to the general area of isoperimetric inequalities are estimates for quantities of physical origin (moments of inertia, torsional rigidity of an elastic beam, fundamental frequency of a membrane, electrostatic capacity, etc.) in terms of geometrical characteristics. An exact isoperimetric inequality is equivalent to the solution of some extremum problem. Isoperimetric inequalities can connect two or more quantities.

For the best known isoperimetric inequalities, namely the classical one and its analogues in the Minkowski spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i0528604.png" />, the Lobachevskii spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i0528605.png" />, the spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i0528606.png" />, as well as for refinements of them, see [[Isoperimetric inequality, classical|Isoperimetric inequality, classical]].

An extensive coverage of isoperimetric inequalities between the elements of the simplest figures, mainly polygons, can be found in [[#References|[1]]]. Such isoperimetric inequalities are called geometric inequalities.

For elementary isoperimetric inequalities between such parameters of sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i0528607.png" /> as the volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i0528608.png" />, the diameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i0528609.png" /> and the radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286010.png" /> of the smallest circumscribed ball, etc., see [[#References|[2]]] and [[#References|[3]]]. Among them are: Young's inequality:

<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/i052860/i05286011.png" /></td> </tr></table>

Gale's inequality:

<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/i052860/i05286012.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286013.png" /> is the length of the edge of the smallest circumscribed regular simplex; Bieberbach's inequality:

<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/i052860/i05286014.png" /></td> </tr></table>

and the Loomis–Whitney inequality:

<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/i052860/i05286015.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286016.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286017.png" />-dimensional volume of the projection of the set onto the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286018.png" />-th of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286019.png" /> pairwise-different <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286020.png" />-dimensional Cartesian coordinate planes. The first three inequalities can be generalized to the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286023.png" /> (see [[#References|[4]]], [[#References|[5]]]). In Bieberbach's inequality, the diameter can be replaced by the mean width (see [[#References|[5]]]).

In connection with problems of [[Arrangement|arrangement]] and [[Covering|covering]], isoperimetric inequalities are considered that are specific for polyhedra, with the introduction of the number of edges or the sum of their lengths, etc. (see [[#References|[6]]]).

For convex bodies, many isoperimetric inequalities (including the classical ones and a number of inequalities between integrals of symmetric functions of principal curvatures) are special cases of inequalities between composite objects (see [[Mixed-volume theory|Mixed-volume theory]]; [[Minkowski inequality|Minkowski inequality]]).

The use of isoperimetric inequalities as estimates for some parameters of figures in terms of others arose within the limits of geometry. The class of isoperimetric inequalities is enriched by mathematical physics, the theory of functions of a complex variable, functional analysis, the theory of approximations of functions, and the calculus of variations. Isoperimetric inequalities in Riemannian geometry are noticeably more complex.

In mathematical physics, isoperimetric inequalities arose (firstly as conjectures) in papers of A. Saint-Venant (1856):

<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/i052860/i05286024.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286025.png" /> is the torsional rigidity of a prismatic elastic beam; of Lord Rayleigh (1877):

<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/i052860/i05286026.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286027.png" /> is the fundamental frequency of the membrane and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286028.png" /> is the first positive root of the Bessel function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286029.png" />; and in papers of H. Poincaré (1903):

<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/i052860/i05286030.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286031.png" /> is the electrostatic capacity of the body. In these inequalities, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286032.png" /> is, respectively, the cross-sectional area of the beam, the area of the membrane and the volume of the body. Numerous results of this sort are summarized in [[#References|[7]]] and [[#References|[8]]]. Certain estimates for the first eigen value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286033.png" /> of the Laplace operator on closed Riemannian manifolds can be found in [[#References|[9]]].

In functional analysis, conditions for boundedness and compactness of imbedding operators (see [[Imbedding theorems|Imbedding theorems]]) for Sobolev spaces have been given in terms of isoperimetric inequalities (connecting measure and capacity) (see [[#References|[10]]], [[#References|[11]]]).

For example, the estimate

<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/i052860/i05286034.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286035.png" /> is a non-negative measure, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286037.png" />, is valid for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286038.png" /> if and only if the following isoperimetric inequality is satisfied for all compacta <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286039.png" />:

<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/i052860/i05286040.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286041.png" /> is the Wiener capacity (see [[Capacity|Capacity]]).

Isoperimetric inequalities for volume and area are used in the proof of a priori estimates for solutions of linear and quasi-linear elliptic equations (see [[#References|[12]]], [[#References|[26]]]).

Specific isoperimetric inequalities arise for convex bodies in a Minkowski space in connection with the theory of approximation of functions (see [[Self-perimeter|Self-perimeter]]; [[Width|Width]]).

Applying isoperimetric inequalities is a standard device in the theory of conformal and quasi-conformal mappings. Inequality (4) below is an example of a conformally-invariant isoperimetric inequality.

Isoperimetric inequalities involving the mean curvature of a submanifold, in particular for minimal surfaces, play an important part in the solution of the [[Plateau problem|Plateau problem]].

In the Riemannian geometry of non-homogeneous spaces, generalizations of the classical isoperimetric inequalities have been studied in detail only in the two-dimensional case. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286042.png" /> be a simply-connected compact two-dimensional manifold with boundary such that the positive part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286043.png" /> of the integral curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286044.png" /> is less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286045.png" />. Then (see [[#References|[13]]]):

<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/i052860/i05286046.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

The isoperimetric inequality (1) is valid also for a [[Two-dimensional manifold of bounded curvature|two-dimensional manifold of bounded curvature]], which is a more general type of manifold than a Riemannian manifold. Equality in (1) is attained for a non-regular object — a domain isometric to the lateral surface of a right circular cone with complete angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286047.png" /> about the vertex. Using (1), inequalities can be established for the length of a curve, inside the domain, in dependence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286049.png" /> and the proper rotation (cf. [[Winding number|Winding number]]). In particular, for the geodesic length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286050.png" />,

<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/i052860/i05286051.png" /></td> </tr></table>

<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/i052860/i05286052.png" /></td> </tr></table>

Isoperimetric inequality (1) is a special case of the estimate

<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/i052860/i05286053.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286054.png" /> is any real number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286055.png" /> is the [[Euler characteristic|Euler characteristic]] of the compact domain with boundary, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286056.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286057.png" /> is the [[Gaussian curvature|Gaussian curvature]]. Similar to (2) are estimates for the area of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286058.png" />-neighbourhood of the boundary of a domain and for the largest distance of points in the domain from the boundary (see [[#References|[14]]]). If the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286059.png" /> is a smooth submanifold in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286060.png" />, then the estimates (1), (2) are supplemented by isoperimetric inequalities involving the external characteristics of the surface. For closed surfaces, the following sharp isoperimetric inequality follows from integral identities (see [[#References|[15]]]):

<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/i052860/i05286061.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286062.png" /> is the radius of a ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286063.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286064.png" />. Similar (but non-sharp) inequalities have been obtained also for surfaces with boundary (see [[#References|[16]]]). In particular, for a simply-connected saddle surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286065.png" /> with boundary length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286066.png" />:

<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/i052860/i05286067.png" /></td> </tr></table>

<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/i052860/i05286068.png" /></td> </tr></table>

The inequalities mentioned remain true for general (non-regular) surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286070.png" />, if instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286071.png" /> the positive exterior curvature is taken into consideration — a measure in the set of locally supporting planes (see [[#References|[16]]]).

For an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286072.png" />-dimensional Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286073.png" />, isoperimetric inequalities are usually connected with one-sided bounds on the [[Sectional curvature|sectional curvature]] or the [[Ricci curvature|Ricci curvature]]. The simplest is a bound for the volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286074.png" /> of the ball of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286075.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286076.png" /> in terms of the volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286077.png" /> of a ball of the same radius in a complete simply-connected surface of constant curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286078.png" />:

<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/i052860/i05286079.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286080.png" /> is the smallest value of the Ricci curvature in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286081.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286082.png" />, then it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286083.png" /> (see [[#References|[17]]]). Similar isoperimetric inequalities are valid for a tubular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286084.png" />-neighbourhood of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286085.png" />-dimensional submanifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286087.png" />; the minimum of the sectional curvatures in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286088.png" /> and the maximum of the normal curvatures of submanifolds participate in such an isoperimetric inequality (instead of the Ricci curvature) (see [[#References|[18]]]).

If the least upper bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286089.png" /> of the sectional curvatures is negative, then the volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286090.png" /> of the closed manifold is bounded from below by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286091.png" /> (see [[#References|[19]]]). The following linear isoperimetric inequality holds for a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286092.png" /> in a complete simply-connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286093.png" />:

<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/i052860/i05286094.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286095.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286096.png" />-dimensional area of the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286097.png" />, and also the isoperimetric inequality

<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/i052860/i05286098.png" /></td> </tr></table>

the exact value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i05286099.png" /> being unknown.

In spaces of negative curvature, a number of estimates have been obtained for convex domains that generalize isoperimetric inequalities for convex bodies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860100.png" /> (see [[#References|[20]]], [[#References|[21]]]). Thus,

<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/i052860/i052860101.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860102.png" /> is the greatest distance of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860103.png" /> from the boundary. If the greatest lower bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860104.png" /> of the sectional curvatures in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860105.png" /> is positive, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860106.png" />, and the left inequality can be sharpened:

<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/i052860/i052860107.png" /></td> </tr></table>

The inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860108.png" /> holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860109.png" /> is the integral [[Mean curvature|mean curvature]]. The following isoperimetric inequality holds in the three-dimensional case:

<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/i052860/i052860110.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860111.png" /> is the Euler characteristic of the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860112.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860113.png" /> is the integral [[Scalar curvature|scalar curvature]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860114.png" />.

In the classical isoperimetric inequality, the area is bounded from above. For closed simply-connected surfaces, the area can be bounded from below in terms of the length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860115.png" /> of the shortest loop that is not homotopic to zero:

<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/i052860/i052860116.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

The exact value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860117.png" /> is known only for the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860118.png" /> and the projective plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860119.png" />. Inequality (4) is a corollary of the following isoperimetric inequality (see [[#References|[22]]]):

<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/i052860/i052860120.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4prm)</td></tr></table>

for the extremal length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860121.png" /> of the family of loops that are not homotopic to zero.

The problem of similar inequalities for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860122.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860123.png" /> is discussed in [[#References|[22]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860124.png" /> is a topological <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860125.png" />-dimensional cube with interior metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860126.png" />, then its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860127.png" />-volume satisfies the inequality

<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/i052860/i052860128.png" /></td> </tr></table>

here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860129.png" /> is the distance in the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860130.png" /> between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860131.png" />-th pair of opposite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860132.png" />-dimensional faces. See [[#References|[24]]], [[#References|[25]]] for more details.

In the theory of minimal surfaces and surfaces like them, a number of isoperimetric inequalities have been obtained that hold not only for smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860133.png" />-dimensional submanifolds in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860134.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860135.png" />, but also for more general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860136.png" />-dimensional "films" : submanifolds with singularities, currents, etc. Thus, the following inequality was established in [[#References|[26]]], [[#References|[27]]]:

<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/i052860/i052860137.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860138.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860139.png" />-area of a face and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860140.png" /> is the integral of the modulus of the mean curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860141.png" /> of the film. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860142.png" />, then the following isoperimetric inequality holds when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860143.png" />:

<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/i052860/i052860144.png" /></td> </tr></table>

In methods of proof and applications, lower bounds for the volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860145.png" /> of the intersection of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860146.png" />-film <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860147.png" /> with the ball of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860148.png" /> and centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860149.png" /> belong to the class of inequalities of the type (5). Thus, for a minimal surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860150.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860151.png" /> increases for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860152.png" />. See [[#References|[27]]], [[#References|[28]]] for some generalizations to minimal films in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860153.png" /> (and to films under the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860154.png" />).

====References====
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====Comments====
For an up-to-date account of geometric inequalities supplementing [[#References|[1]]], see [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Bandle, "Isoperimetric inequalities and applications" , Pitman (1980) {{MR|0572958}} {{ZBL|0436.35063}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Kaul, "Isoperimetrische Ungleichung und Gauss–Bonnet–Formel für <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052860/i052860158.png" />-Flächen in Riemannsche Mannigfaltigkeiten" ''Arch. Rat. Mech. Anal.'' , '''45''' (1972) pp. 194–221  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D.S. Mitrinović, J.E. Pečarić, V. Volenec, "Recent advances in geometric inequalities" , Kluwer (1989) {{MR|1022443}} {{ZBL|0679.51004}} </TD></TR></table>