n-dimensional isoperimetric inequality

Isoperimetric inequalitiesMathworldPlanetmath for 2 and 3 dimensionsPlanetmathPlanetmath 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.


We shall use cartesian coordinatesMathworldPlanetmath based on the ortho-normal vector system B¯1,B¯2, etc… An (n-1) surface S is defined by a functionMathworldPlanetmath of n coordinatesMathworldPlanetmathPlanetmath equated to zero. On this surface, any coordinate can be considered as a function of all the others. Let us take the last one xn, as a function of x1,x2xn-1, 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 infinitesimalMathworldPlanetmath surface element ds is (see the annex):


Zi are the partial derivativesMathworldPlanetmath of z with respect to xi. This surface constraint is handled with the help of a Lagrange multiplier R which allows us to maximize a single function F:


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:


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


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.


The volume BVn of an n-ball of radius R is (ref 1):


Γ is Euler’s gamma functionDlmfDlmfMathworldPlanetmath. Since this volume is obviously the integralDlmfPlanetmath of the surface from 0 to R, the surface is the derivative of the volume with respect to R:


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


This equality holds for an n-ball. The volume Vn of an arbitrary solid of area An cannot be greater than the volume BVn of an n-ball with the same area; therefore the following inequalityMathworldPlanetmath holds:


This is the so-called isoperimetric inequality for n dimensions.


The infinitesimal surface element of an n-dimensional solid is in fact the volume of an infinitesimal (n-1)-parallelepipedMathworldPlanetmath. This volume (ref 2) is the square rootMathworldPlanetmath of the Gram determinantMathworldPlanetmath of the edge vectors U¯1,U¯2U¯n-1. The elements of this determinantDlmfMathworldPlanetmath are the dot productsMathworldPlanetmath of the edge vectors:


Let P be the position vector of a point in the (n-1) dimensional enveloppe of the solid. δ¯i is the infinitesimal displacement we get by varying the coordinate xi by dxi and keeping all the other (n-2) independent variables fixed. Only the last coordinate z varies to maintain the new position into the envelope:


Zi is a shortcut for the partial derivative of z with respect to xi. The (n-1) infinitesimal vectors δ¯i define an (n-1)-parallelepiped and its Gram determinant is:


δij is the Kronecker symbolMathworldPlanetmath. In the determinant, dxi appears in one row and one column, so that it can be factored out twice. Therefore, the volume of the (n-1)-parallelepiped δ¯i, or the surface element ds is:


The determinant of the matrix H defined by Hij=δij+ZiZj is the product of its eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath. For any (n-1)-vector W¯ we have:


Z¯ being the (n-1)-vector (Z1,Z2Zn-1). If W¯ is orthogonalMathworldPlanetmathPlanetmath to Z¯, HW¯=W¯ and its eigenvalue is 1. But there are (n-2) such vectors, so the determiant is the last eignevalue for W¯=Z¯: HZ¯=(1+|Z|2)Z¯. Finally:



  • 1 Eric Weinstein - HypersphereMathworldPlanetmath
    An elegant proof of the hypersphere volume formula.
  • 2 Nils Barth - The Gramian and K-Volume in N-Space
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