Isoperimetric inequality

The isoperimetric inequality is a mathematical inequality from geometry that estimates the area of a figure against its circumference in a plane and the volume of a body against its surface area in three-dimensional space . At the same time, it characterizes a special position of the circle among all figures in the plane as well as a special position of the sphere among all bodies in three-dimensional space, which consists in the fact that the case of equality in this inequality occurs only in the case of the circle or the sphere.

This means that among all figures in the plane with the same circumference, the circle includes the largest area, and accordingly that of all bodies in three-dimensional space with the same surface, the sphere has the largest volume. The circle in the plane and the sphere in space are solutions to the isoperimetric problem (to find a closed curve that encloses the largest content for a given circumference).

The analogous statement also applies in -dimensional Euclidean space : Among all bodies with the same -dimensional surface area, the -dimensional sphere has the largest -dimensional volume. ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle (n-1)}$ ${\displaystyle n}$ ${\displaystyle n}$

The word isoperimetric comes from the Greek: iso stands for “equal” and perimeter means “circumference”.

Figures in the plane

Among all partial areas of the two-dimensional plane with finite extent and a well-defined circumference, the circle has the property that it includes the largest area for a given circumference. This fact can be represented by the isoperimetric inequality of the plane : ${\displaystyle \mathbb {R} ^{2}}$

${\displaystyle A\leq {\frac {U^{2}}{4\pi }}\,,}$

where represents the perimeter of the area and the enclosed area. Equality occurs if and only if the geometrical figure under consideration is either the circle itself or the circle after an insignificant modification which leaves the circumference and the area unchanged (e.g. by removing the center of the circle). ${\displaystyle U\,\!}$ ${\displaystyle A\,\!}$

Body in three-dimensional space

Among all sub-volumes of three-dimensional space with finite dimensions and a well-defined surface area , the sphere has the comparable property that it encloses the largest volume for a given surface area . The isoperimetric inequality of three-dimensional space is: ${\displaystyle \mathbb {R} ^{3}}$

${\displaystyle V\leq {\frac {A^{3/2}}{6\pi ^{1/2}}}\,,}$

where describes the surface area and the enclosed volume. Here, too, equality occurs precisely when the geometrical body in question is either the sphere itself, or when it is an insignificant modification of the same, which leaves the surface area and volume unchanged (for example a sphere after removing some points located within the sphere or after removing a one-dimensional line running through the sphere). ${\displaystyle A\,\!}$ ${\displaystyle V\,\!}$

n-dimensional bodies in n-dimensional space with n ≥ 2

For the general formulation, it is advisable to use the -dimensional Lebesgue measure , which assigns its -dimensional volume to each area in the -dimensional volume , and the ( n -1) -dimensional Hausdorff measure , which assigns a measure to the topological edge of a - dimensional rectifiable edge corresponds to the heuristic -dimensional surface content. ${\displaystyle n}$ ${\displaystyle L^{n}\,\!}$ ${\displaystyle G\,\!}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle n}$ ${\displaystyle L^{n}(G)\,\!}$ ${\displaystyle H^{n-1}\,\!}$ ${\displaystyle \partial G}$ ${\displaystyle G\,\!}$ ${\displaystyle H^{n-1}(\partial G)}$ ${\displaystyle (n-1)}$ ${\displaystyle (n-1)}$

For every non-empty restricted area in the with a rectifiable boundary holds ${\displaystyle G\,\!}$ ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle {\frac {L^{n}(G)^{1/n}}{H^{n-1}(\partial G)^{1/(n-1)}}}\leq {\frac {L^{n}(B)^{1/n}}{H^{n-1}(\partial B)^{1/(n-1)}}}\,,}$

where (English for "ball") stands for any n -dimensional ball im . The right side of the inequality is independent of the radius of the sphere (> 0) and of its center in . ${\displaystyle B\,\!}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$

Equality occurs if and only if such an n -dimensional sphere is im (or a modification of one which leaves the Lebesgue measure of the area and Hausdorff measure of its edge unchanged). ${\displaystyle G\,\!}$ ${\displaystyle \mathbb {R} ^{n}}$

For and one gets back the formulations made above. ${\displaystyle n=2}$ ${\displaystyle n=3}$

m-dimensional minimal areas in n-dimensional space with 2 ≤ m <n

Here we consider compactly bounded -dimensional minimal areas in (with ), these are -dimensional areas with the property that they have the smallest area for a given fixed ( dimensional) edge (compact). Always applies ${\displaystyle m}$ ${\displaystyle M}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle 2\leq m<n}$ ${\displaystyle m}$ ${\displaystyle m-1}$ ${\displaystyle \partial M}$

${\displaystyle {\frac {H^{m}(M)^{1/m}}{H^{m-1}(\partial M)^{1/(m-1)}}}\leq {\frac {L^{m}(B)^{1/m}}{H^{m-1}(\partial B)^{1/(m-1)}}}\,,}$

where the letter stands for -dimensional spheres. In the case of equality, there is even a -dimensional sphere (and the edge of a -dimensional circle) located in a -dimensional subplane of the . ${\displaystyle B}$ ${\displaystyle m}$ ${\displaystyle M}$ ${\displaystyle m}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle m}$ ${\displaystyle M}$ ${\displaystyle (m-1)}$

The fact that one has to limit oneself to minimal areas in the formulation becomes plausible when one considers that an area spanned by a fixed edge can be enlarged without affecting its edge. Think of a soap film (with a closed wire as a fixed edge), which gains in surface area through a stream of air (while maintaining the edge).

Reverse isoperimetric inequalities

Obviously, there cannot be an immediate reverse of the isoperimetric inequality. As an example, consider those figures with a fixed area in the plane: you can realize any size you want by making a rectangle "as thin as you want". The length of one side then tends to zero, and so, with the area remaining the same, the other must taper to infinity. However, there are (limited) reversals. These are:

(1) Let be a convex body in and a -dimensional regular simplex ( -gon; triangle for and tetrahedron for ). Then there is an affine image of with the property ${\displaystyle K}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle T}$ ${\displaystyle n}$ ${\displaystyle n+1}$ ${\displaystyle n=2}$ ${\displaystyle n=3}$ ${\displaystyle K}$ ${\displaystyle K}$

${\displaystyle V(K')=V(T{\big )}}$ and ${\displaystyle S(K')\leq S(T).}$

(2) If there is a symmetrical convex body in and a -dimensional cube, then there is an affine image of such that ${\displaystyle K}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle W}$ ${\displaystyle n}$ ${\displaystyle K^{\prime }}$ ${\displaystyle K}$

${\displaystyle V(K^{\prime })=V(W{\big )}}$ and ${\displaystyle S(K^{\prime })\leq S(W).}$

(Here = volume and = surface) ${\displaystyle V}$ ${\displaystyle S}$

With the exception of affine images, the simplexes or the cuboids have the maximum surface area of ​​all convex (symmetrical) bodies for a given volume. (In other words: among the reference bodies mentioned here, simplex or cuboid have the "worst" isoperimetric ratio and not the "best" like the sphere.) ${\displaystyle S/V}$

These reverse isoperimetric inequalities are from Keith Ball and their proofs are based on theorems of Fritz John and Brascamp / Lieb .

The problem of Dido

Occasionally the term problem of dido appears in connection with the isoperimetric inequality. According to tradition, the Phoenician Queen Dido was allowed to mark out a piece of land for her people with a cowhide when the city of Carthage was founded. After the fur had been cut into thin strips and these strips were sewn together to form a ribbon, the question arose as to which geometric shape the territory bounded by this ribbon should have so that its area would take on a maximum.

In comparison to the isoperimetric inequality in the plane, two peculiarities arise with this question:

1. The land to be staked was on the coast (which we will assume as a straight line for the sake of simplicity). This would have to be replaced by a half-plane, the edge of which, as a “ supporting line ”, already accommodates part of the edge of the area sought, while the cowhide describes the remaining edge that can be freely shaped. ${\displaystyle \mathbb {R} ^{2}}$
2. The earth is a sphere. That would have to be replaced by a (large) spherical surface or the half-plane by a hemisphere. ${\displaystyle \mathbb {R} ^{2}}$

To solve the first problem, the band is laid out in a semicircle line so that its ends come to rest on the supporting straight line. After considering symmetry in the two-dimensional case, the semicircle has the largest area in the half-plane with free edge length in the half-plane edge and a given fixed edge length in the interior of the half-plane.

literature

• Isaac Chavel : Isoperimetric Inequalities: Differential geometric and analytic perspectives , Cambridge University Press, 2001
• Juri Dmitrijewitsch Burago , Wiktor Abramowitsch Salgaller Geometric inequalities , Springer 1988
• Robert Osserman : The isoperimetric inequality. In: Bulletin of the American Mathematical Society. Vol. 84, No. 6, 1978, ISSN  0273-0979 , pp. 1182-1238, online .

swell

1. ^ Frederick Almgren : Optimal isoperimetric inequalities. In: Indiana University Mathematics Journal. Vol. 35, 1986, ISSN  0022-2518 , pp. 451-547.
2. ^ Keith Ball : Volume ratios and a reverse isoperimetric inequality .