Isoperimetric inequality
Isoperimetric the inequality is a mathematical inequality from the geometry, the surface area of the figure to its circumference in the plane of and estimate in three dimensional space, the volume of a body to its surface area. At the same time it characterizes a special position of the circle under all the figures in the plane and a special position of the ball among all bodies in three-dimensional space, which is that only the circuit or in the sphere of equality occurs in this inequality.
This means that includes all of the figures with the same extent in the plane of the circle of the largest surface area, and accordingly, that of all the objects in the three-dimensional space with the same surface of the sphere has the largest volume. The circle in the plane and the sphere in space are solutions of the isoperimetric problem ( to find a closed curve that encloses most of the content for a given scope ).
Also in n-dimensional Euclidean space, the analogous statement is true: Among all bodies with the same ( n-1) -dimensional surface area of the n- dimensional sphere has the largest n- dimensional volume.
The word isoperimetric derived from the Greek: iso stands for " equal ", and perimeter means " scale".
Figures in the plane
In all partial areas of the two-dimensional plane having a finite extent and a well-defined scope of the circuit has the characteristic that it includes the largest surface area for a given scale. This situation can be represented by the isoperimetric inequality of the level:
Where the extent of the area represented and the enclosed area. Equality holds if and only when the considered geometric figure either the circle itself, or the circle after a minor modification which leaves unchanged the scope and content area (for example, by removal of the circle center ).
A body in three-dimensional space
In all sub-volumes of the three-dimensional space having a finite extent, and a well-defined surface content of the sphere has the similar property that it includes the largest volume for a given surface area. The isoperimetric inequality of the three-dimensional space is:
Where the surface area and describes the enclosed volume. Here, too, equality if and when the considered geometrical body, either the ball itself, or if it is a minor modification is the same, which surface area and volume can be changed ( for example, a ball after removing some within the Kugel points located or after removal of a plane running through the ball -dimensional distance).
N-dimensional body in the n- dimensional space with n ≥ 2
For general formulation is appropriately used as the n-dimensional Lebesgue measure, which assigns each area in its n - dimensional volume, and the (n-1 )-dimensional Hausdorff measure, which assigns to the topological boundary of a measure which, in case an (n -1 )-dimensional rectifiable edge corresponds to the heuristic (n-1 )-dimensional surface area.
For every non-empty bounded area in the border with rektifizierbarem
Where ( English for " ball" ) stands for an arbitrary n- dimensional sphere in. The right side of the inequality is independent of the radius of the sphere (> 0 ) and the center of which in the.
Equality occurs if and when even such a n-dimensional ball is in (or modification of such, which Lebesgue measure of the area and Hausdorff measure of its edge leaves unchanged ).
For n = 2 and n = 3, it is returned to the formulations made above.
M-dimensional minimum areas in n-dimensional space with 2 ≤ m
Here we consider compact bounded m-dimensional minimal surfaces M in ( with 2 ≤ m < n) are m-dimensional space with the property that they possess at a given solid (m-1 -dimensional ) boundary (compact) the smallest area.
Always apply
In which case the letter B stands for m- dimensional spheres.
In the equality case M is even one in a m-dimensional sub-plane of the preferred m- dimensional sphere ( and the boundary of M is an m-1 -dimensional circle).
That one in the formulation must be limited to minimal surfaces, is plausible, considering that one out of a sturdy rim spanned area can be increased, without compromising their edge is affected.
You think of a soap film ( with a closed wire as a solid boundary ), added wins by a flow of air to surface area (while maintaining the edge ).
Reverse isoperimetric inequalities
Obviously, there can be no direct reversal of the isoperimetric inequality.
As an example, consider those figures in the plane with a fixed surface area: one can realize an arbitrarily large extent by " arbitrarily thin " makes about a rectangle.
The page length then goes to zero, and as a consequence, at constant area, the others go to infinity.
Nevertheless, there are (limited ) reversals.
These are:
( Here V is the volume and S = surface, surface )
Up to affine transformations so under all convex ( symmetric ) bodies have the simplexes or cubes for a given volume, the maximum surface.
( In other words, among the reference blocks mentioned here have simplex or cube the "worst" isoperimetric ratio and not the " best" as the sphere. )
This reverse isoperimetric inequalities are those of Keith Ball and their proofs are based on sets of Fritz John and Brascamp / Lieb.
The problem of Dido
Appears occasionally the term problem of Dido in connection with the isoperimetric inequality.
According to tradition, the Phoenician Queen Dido was allowed to stake out a cowhide with a piece of land for their people in the founding of the city of Carthage.
After the fur was split into thin strips and these strips were sewn together to form a band, the question which the geometric shape bounded by this band territory should have now, so that its surface assumes a maximum set.
Compared to the isoperimetric inequality in the plane occur with this issue two special features:
To solve the first problem, it sets up the tape to form a half circular line such that the ends of which come to lie on the supporting line.
Because after a symmetry consideration in the two-dimensional case has in the half-plane of the semicircle with free edge length in the half-plane edge and predetermined fixed edge length inside the half-plane, the largest area.
Here we consider compact bounded m-dimensional minimal surfaces M in ( with 2 ≤ m < n) are m-dimensional space with the property that they possess at a given solid (m-1 -dimensional ) boundary (compact) the smallest area. Always apply
In which case the letter B stands for m- dimensional spheres. In the equality case M is even one in a m-dimensional sub-plane of the preferred m- dimensional sphere ( and the boundary of M is an m-1 -dimensional circle).
That one in the formulation must be limited to minimal surfaces, is plausible, considering that one out of a sturdy rim spanned area can be increased, without compromising their edge is affected. You think of a soap film ( with a closed wire as a solid boundary ), added wins by a flow of air to surface area (while maintaining the edge ).
Reverse isoperimetric inequalities
Obviously, there can be no direct reversal of the isoperimetric inequality. As an example, consider those figures in the plane with a fixed surface area: one can realize an arbitrarily large extent by " arbitrarily thin " makes about a rectangle. The page length then goes to zero, and as a consequence, at constant area, the others go to infinity. Nevertheless, there are (limited ) reversals. These are:
( Here V is the volume and S = surface, surface )
Up to affine transformations so under all convex ( symmetric ) bodies have the simplexes or cubes for a given volume, the maximum surface. ( In other words, among the reference blocks mentioned here have simplex or cube the "worst" isoperimetric ratio and not the " best" as the sphere. )
This reverse isoperimetric inequalities are those of Keith Ball and their proofs are based on sets of Fritz John and Brascamp / Lieb.
The problem of Dido
Appears occasionally the term problem of Dido in connection with the isoperimetric inequality. According to tradition, the Phoenician Queen Dido was allowed to stake out a cowhide with a piece of land for their people in the founding of the city of Carthage. After the fur was split into thin strips and these strips were sewn together to form a band, the question which the geometric shape bounded by this band territory should have now, so that its surface assumes a maximum set.
Compared to the isoperimetric inequality in the plane occur with this issue two special features:
To solve the first problem, it sets up the tape to form a half circular line such that the ends of which come to lie on the supporting line. Because after a symmetry consideration in the two-dimensional case has in the half-plane of the semicircle with free edge length in the half-plane edge and predetermined fixed edge length inside the half-plane, the largest area.