# 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 $x_{1},x_{2}...x_{n-1},$ and call it z for brevity. This surface is the envelope of an n-dimensional solid of volume V:

 $V=\int_{V}dx_{1}dx_{2}...dz=\int_{S}zdx_{1}dx_{2}...dx_{n-1}$

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

 $S=\int_{S}ds=A$

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

 $S=\int_{S}\sqrt{1+Z_{1}^{2}+...+Z_{n-1}^{2}}dx_{1}...dx_{n-1}$

$~{}~{}~{}~{}$$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:

 $I=\int_{S}Fdx_{1}...dx_{n-1}=\int_{S}\left(z+R\sqrt{1+Z_{1}^{2}+...+Z_{n-1}^{2% }}\right)dx_{1}...dx_{n-1}$

$~{}~{}~{}~{}$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:

 $\frac{\partial}{\partial{x_{i}}}\left(\frac{Z_{i}}{\sqrt{1+Z_{1}^{2}+...+Z_{n-% 1}^{2}}}\right)=\frac{1}{R}$

$~{}~{}~{}~{}$After a first integration, and squaring, we have:

 $\frac{Z_{i}^{2}}{1+Z_{1}^{2}+...+Z_{n-1}^{2}}=\frac{(x_{i}-a_{i})^{2}}{R^{2}}$

$~{}~{}~{}~{}$Summing all these equations together, after some algebra, we get:

 $Z_{i}=\frac{\partial{z}}{\partial{x_{i}}}=\frac{x_{i}-a_{i}}{\sqrt{R^{2}-(x_{1% }-a_{1})^{2}-...-(x_{n-1}-a_{n-1})^{2}}}$

$~{}~{}~{}~{}$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 $BV_{n}$ of an n-ball of radius R is (ref 1):

 $BV_{n}=\frac{\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}+1\right)}R^{n}$

$~{}~{}~{}~{}\Gamma$ 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:

 $BA_{n}=\frac{2\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}\right)}R^{n-1}$

$~{}~{}~{}~{}~{}$Eliminating the radius R between these two equations, we get:

 $\frac{\left(BV_{n}\right)^{n-1}}{\left(BA_{n}\right)^{n}}=\frac{\pi^{-\frac{n}% {2}}\left[\Gamma\left(\frac{n}{2}\right)\right]^{n}}{2^{n}\left[\frac{n}{2}% \Gamma\left(\frac{n}{2}\right)\right]^{n-1}}=\frac{\Gamma\left(\frac{n}{2}+1% \right)}{\left(n\sqrt{\pi}\right)^{n}}$

$~{}~{}~{}~{}$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 $BV_{n}$ of an n-ball with the same area; therefore the following inequality holds:

 $\frac{\left(V_{n}\right)^{n-1}}{\left(A_{n}\right)^{n}}\leq{\frac{\Gamma\left(% \frac{n}{2}+1\right)}{\left(n\sqrt{\pi}\right)^{n}}}$

$~{}~{}~{}~{}$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 $\overline{U}_{1},\overline{U}_{2}...\overline{U}_{n-1}$. The elements of this determinant are the dot products of the edge vectors:

 $G_{ij}=\overline{U}_{i}\cdotp\overline{U}_{j}$

$~{}~{}~{}~{}~{}$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 $dx_{i}$ and keeping all the other (n-2) independent variables fixed. Only the last coordinate z varies to maintain the new position into the envelope:

 $\overline{\delta}_{i}=dx_{i}\overline{B}_{i}+dz\overline{B}_{n}=dx_{i}(% \overline{B}_{i}+\frac{\partial{z}}{\partial{x_{i}}}\overline{B}_{n})=dx_{i}(% \overline{B}_{i}+Z_{i}\overline{B}_{n})$

$~{}~{}~{}~{}~{}$$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:

 $G_{ij}=\overline{\delta}_{i}\cdotp\overline{\delta}_{j}=(\delta_{ij}+Z_{i}Z_{j% })dx_{i}dx_{j}$

$~{}~{}~{}~{}~{}$$\delta_{ij}$ is the Kronecker symbol. In the determinant, $dx_{i}$ 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:

 $ds=\sqrt{\|\delta_{ij}+Z_{i}Z_{j}\|}dx_{1}dx_{2}...dx_{n-1}$

$~{}~{}~{}~{}~{}$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:

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

$~{}~{}~{}~{}~{}$$\overline{Z}$ being the (n-1)-vector $(Z_{1},Z_{2}...Z_{n-1})$. 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:

 $ds=\sqrt{Z_{1}^{2}+Z_{2}^{2}+...+Z_{n-1}^{2}}dx_{1}dx_{2}...dx_{n-1}$

## References

• 1 Eric Weinstein - http://mathworld.wolfram.com/Hypersphere.html
An elegant proof of the hypersphere volume formula.
• 2 Nils Barth - The Gramian and K-Volume in N-Space
http://www.jyi.org/volumes/volume2/issue1/articles/barth.html
Title n-dimensional isoperimetric inequality NdimensionalIsoperimetricInequality 2013-03-22 19:20:01 2013-03-22 19:20:01 dh2718 (16929) dh2718 (16929) 4 dh2718 (16929) Theorem msc 51M16 msc 51M25