n-dimensional isoperimetric inequality
Isoperimetric inequalities for 2 and 3 dimensions are generalized here to n dimensions. First, the n-dimensional ball is shown to have the greatest volume for a given (n-1)-surface area. Then, the volume and area of the n-ball are used to establish the n-dimensional isoperimetric inequality.
1 THE GREATEST N-VOLUME
We shall use cartesian coordinates based on the ortho-normal vector system etc… An (n-1) surface S is defined by a function of n coordinates equated to zero. On this surface, any coordinate can be considered as a function of all the others. Let us take the last one , as a function of and call it z for brevity. This surface is the envelope of an n-dimensional solid of volume V:
We are going to maximize V, subject to the condition that the surface S has a given area A:
The infinitesimal surface element is (see the annex):
The solution to this variational problem is given by the n-1 Euler-Lagrange equations:
In our case, they turn to be:
After a first integration, and squaring, we have:
This system is easily integrated and gives exactly the equation on an n-ball, which is therefore a stationary point of the functional I. Since the minimum volume is obviously zero for a flat solid, the n-ball has necessarily the maximum volume.
2 THE ISOPERIMETRIC INEQUALITY
The volume of an n-ball of radius R is (ref 1):
Eliminating the radius R between these two equations, we get:
This is the so-called isoperimetric inequality for n dimensions.
3 ANNEX: N-DIMENSIONAL PARALLELEPIPED
The infinitesimal surface element of an n-dimensional solid is in fact the volume of an infinitesimal (n-1)-parallelepiped. This volume (ref 2) is the square root of the Gram determinant of the edge vectors . The elements of this determinant are the dot products of the edge vectors:
Let P be the position vector of a point in the (n-1) dimensional enveloppe of the solid. is the infinitesimal displacement we get by varying the coordinate by and keeping all the other (n-2) independent variables fixed. Only the last coordinate z varies to maintain the new position into the envelope:
is a shortcut for the partial derivative of z with respect to . The (n-1) infinitesimal vectors define an (n-1)-parallelepiped and its Gram determinant is:
is the Kronecker symbol. In the determinant, appears in one row and one column, so that it can be factored out twice. Therefore, the volume of the (n-1)-parallelepiped , or the surface element ds is:
The determinant of the matrix H defined by is the product of its eigenvalues. For any (n-1)-vector we have:
being the (n-1)-vector . If is orthogonal to , and its eigenvalue is 1. But there are (n-2) such vectors, so the determiant is the last eignevalue for : . Finally:
Eric Weinstein - Hypersphere
An elegant proof of the hypersphere volume formula.
Nils Barth - The Gramian and K-Volume in N-Space
|Title||n-dimensional isoperimetric inequality|
|Date of creation||2013-03-22 19:20:01|
|Last modified on||2013-03-22 19:20:01|
|Last modified by||dh2718 (16929)|