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 ${\overline{B}}_1,{\overline{B}}_2,$ 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 $x_n$, as a function of $x_1,x_2\mathrm{\dots }{x}_{n-1},$ and call it z for brevity. This surface is the envelope of an n-dimensional solid of volume V:

$$V={\int }_V𝑑x_1𝑑x_2\mathrm{\dots }𝑑z={\int }_{S}z𝑑x_1𝑑x_2\mathrm{\dots }𝑑x_{n-1}$$

We are going to maximize V, subject to the condition that the surface S has a given area A:

$$S={\int }_S𝑑s=A$$

The infinitesimal surface element $ds$ is (see the annex):

$$S={\int }_S\sqrt{1+Z_1^2+\mathrm{\dots }+Z_{n-1}^2}𝑑x_1\mathrm{\dots }𝑑x_{n-1}$$

$Z_i$ are the partial derivatives of z with respect to $x_i$. This surface constraint is handled with the help of a Lagrange multiplier R which allows us to maximize a single function F:

$$I={\int }_{S}F𝑑x_1\mathrm{\dots }𝑑x_{n-1}={\int }_S\left(z+R\sqrt{1+Z_1^2+\mathrm{\dots }+Z_{n-1}^2}\right)𝑑x_1\mathrm{\dots }𝑑x_{n-1}$$

The solution to this variational problem is given by the n-1 Euler-Lagrange equations:

$$\frac{\partial F}{\partial z}-\frac{\partial }{\partial x_i}\left(\frac{\partial F}{\partial Z_i}\right)=0$$

In our case, they turn to be:

$$\frac{\partial }{\partial x_i}\left(\frac{Z_i}{\sqrt{1+Z_1^2+\mathrm{\dots }+Z_{n-1}^2}}\right)=\frac{1}{R}$$

After a first integration, and squaring, we have:

$$\frac{Z_i^2}{1+Z_1^2+\mathrm{\dots }+Z_{n-1}^2}=\frac{{(x_i-a_i)}^2}{R^2}$$

Summing all these equations together, after some algebra, we get:

$$Z_i=\frac{\partial z}{\partial x_i}=\frac{x_i-a_i}{\sqrt{R^2-{(x_1-a_1)}^2-\mathrm{\dots }-{(x_{n-1}-a_{n-1})}^2}}$$

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 $B{V}_n$ of an n-ball of radius R is (ref 1):

$$B{V}_n=\frac{{\pi }^{\frac{n}{2}}}{\mathrm{\Gamma }\left(\frac{n}{2}+1\right)}{R}^n$$

$\mathrm{\Gamma }$ is Euler’s gamma function. Since this volume is obviously the integral of the surface from 0 to R, the surface is the derivative of the volume with respect to R:

$$B{A}_n=\frac{2{\pi }^{\frac{n}{2}}}{\mathrm{\Gamma }\left(\frac{n}{2}\right)}{R}^{n-1}$$

Eliminating the radius R between these two equations, we get:

$$\frac{{\left(B{V}_n\right)}^{n-1}}{{\left(B{A}_n\right)}^n}=\frac{{\pi }^{-\frac{n}{2}}{\left[\mathrm{\Gamma }\left(\frac{n}{2}\right)\right]}^n}{2^n{\left[\frac{n}{2}\mathrm{\Gamma }\left(\frac{n}{2}\right)\right]}^{n-1}}=\frac{\mathrm{\Gamma }\left(\frac{n}{2}+1\right)}{{\left(n\sqrt{\pi }\right)}^n}$$

This equality holds for an n-ball. The volume $V_n$ of an arbitrary solid of area $A_n$ cannot be greater than the volume $B{V}_n$ of an n-ball with the same area; therefore the following inequality holds:

$$\frac{{\left(V_n\right)}^{n-1}}{{\left(A_n\right)}^n}\le \frac{\mathrm{\Gamma }\left(\frac{n}{2}+1\right)}{{\left(n\sqrt{\pi }\right)}^n}$$

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 ${\overline{U}}_1,{\overline{U}}_2\mathrm{\dots }{\overline{U}}_{n-1}$. The elements of this determinant are the dot products of the edge vectors:

$$G_{ij}={\overline{U}}_i\cdot {\overline{U}}_j$$

Let P be the position vector of a point in the (n-1) dimensional enveloppe of the solid. ${\overline{\delta }}_i$ is the infinitesimal displacement we get by varying the coordinate $x_i$ by $d{x}_i$ and keeping all the other (n-2) independent variables fixed. Only the last coordinate z varies to maintain the new position into the envelope:

$${\overline{\delta }}_i=d{x}_i{\overline{B}}_i+dz{\overline{B}}_n=d{x}_i({\overline{B}}_i+\frac{\partial z}{\partial x_i}{\overline{B}}_n)=d{x}_i({\overline{B}}_i+Z_i{\overline{B}}_n)$$

$Z_i$ is a shortcut for the partial derivative of z with respect to $x_i$. The (n-1) infinitesimal vectors ${\overline{\delta }}_i$ define an (n-1)-parallelepiped and its Gram determinant is:

$$G_{ij}={\overline{\delta }}_i\cdot {\overline{\delta }}_j=({\delta }_{ij}+Z_i{Z}_j)d{x}_{i}d{x}_j$$

${\delta }_{ij}$ is the Kronecker symbol. In the determinant, $d{x}_i$ appears in one row and one column, so that it can be factored out twice. Therefore, the volume of the (n-1)-parallelepiped ${\overline{\delta }}_i$, or the surface element ds is:

$$ds=\sqrt{\parallel {\delta }_{ij}+Z_i{Z}_j\parallel }d{x}_{1}d{x}_2\mathrm{\dots }d{x}_{n-1}$$

The determinant of the matrix H defined by $H_{ij}={\delta }_{ij}+Z_i{Z}_j$ is the product of its eigenvalues. For any (n-1)-vector $\overline{W}$ we have:

$$H\overline{W}=\overline{W}+(\overline{Z}\cdot \overline{W})\overline{Z}$$

$\overline{Z}$ being the (n-1)-vector $(Z_1,Z_2\mathrm{\dots }{Z}_{n-1})$. If $\overline{W}$ is orthogonal to $\overline{Z}$, $H\overline{W}=\overline{W}$ and its eigenvalue is 1. But there are (n-2) such vectors, so the determiant is the last eignevalue for $\overline{W}=\overline{Z}$: $H\overline{Z}=(1+{|Z|}^2)\overline{Z}$. Finally:

$$ds=\sqrt{Z_1^2+Z_2^2+\mathrm{\dots }+Z_{n-1}^2}d{x}_{1}d{x}_2\mathrm{\dots }d{x}_{n-1}$$

Title	n-dimensional isoperimetric inequality
Canonical name	NdimensionalIsoperimetricInequality
Date of creation	2013-03-22 19:20:01
Last modified on	2013-03-22 19:20:01
Owner	dh2718 (16929)
Last modified by	dh2718 (16929)
Numerical id	4
Author	dh2718 (16929)
Entry type	Theorem
Classification	msc 51M16
Classification	msc 51M25