n-dimensional isoperimetric inequality

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

B

¯

1

,

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

=

∫

V

𝑑

x

1

⁢

𝑑

x

2

⁢

…

⁢

𝑑

z

=

∫

S

z

⁢

𝑑

x

1

⁢

𝑑

x

2

⁢

…

⁢

𝑑

x

n

-

1

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

S

=

∫

S

𝑑

s

=

A

The

infinitesimal

surface element

d

⁢

s

is (see the annex):

S

=

∫

S

1

+

Z

1

2

+

…

+

Z

n

-

1

2

⁢

𝑑

x

1

⁢

…

⁢

𝑑

x

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

=

∫

S

F

⁢

𝑑

x

1

⁢

…

⁢

𝑑

x

n

-

1

=

∫

S

(

z

+

R

⁢

1

+

Z

1

2

+

…

+

Z

n

-

1

2

)

⁢

𝑑

x

1

⁢

…

⁢

𝑑

x

n

-

1

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

∂

⁡

F

∂

⁡

z

-

∂

∂

⁡

x

i

⁢

(

∂

⁡

F

∂

⁡

Z

i

)

=

0

In our case, they turn to be:

∂

∂

⁡

x

i

⁢

(

Z

i

1

+

Z

1

2

+

…

+

Z

n

-

1

2

)

=

1

R

After a first integration, and

squaring

, we have:

Z

i

2

1

+

Z

1

2

+

…

+

Z

n

-

1

2

=

(

x

i

-

a

i

)

2

R

2

Summing

all these equations together, after some

algebra

, we get:

Z

i

=

∂

⁡

z

∂

⁡

x

i

=

x

i

-

a

i

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

B

⁢

V

n

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

B

⁢

V

n

=

π

n

2

Γ

⁢

(

n

2

+

1

)

⁢

R

n

Γ

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

=

2

⁢

π

n

2

Γ

⁢

(

n

2

)

⁢

R

n

-

1

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

(

B

⁢

V

n

)

n

-

1

(

B

⁢

A

n

)

n

=

π

-

n

2

⁢

[

Γ

⁢

(

n

2

)

]

n

2

n

⁢

[

n

2

⁢

Γ

⁢

(

n

2

)

]

n

-

1

=

Γ

⁢

(

n

2

+

1

)

(

n

⁢

π

)

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:

(

V

n

)

n

-

1

(

A

n

)

n

≤

Γ

⁢

(

n

2

+

1

)

(

n

⁢

π

)

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

U

¯

1

,

U

¯

2

⁢

…

⁢

U

¯

n

-

1

. The elements of this

determinant

are the

dot products

of the edge vectors:

G

i

⁢

j

=

U

¯

i

⋅

U

¯

j

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

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:

δ

¯

i

=

d

⁢

x

i

⁢

B

¯

i

+

d

⁢

z

⁢

B

¯

n

=

d

⁢

x

i

⁢

(

B

¯

i

+

∂

⁡

z

∂

⁡

x

i

⁢

B

¯

n

)

=

d

⁢

x

i

⁢

(

B

¯

i

+

Z

i

⁢

B

¯

n

)

Z

i

is a shortcut for the partial derivative of z with respect to

x

i

. The (n-1) infinitesimal vectors

δ

¯

i

define an (n-1)-parallelepiped and its Gram determinant is:

G

i

⁢

j

=

δ

¯

i

⋅

δ

¯

j

=

(

δ

i

⁢

j

+

Z

i

⁢

Z

j

)

⁢

d

⁢

x

i

⁢

d

⁢

x

j

δ

i

⁢

j

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

δ

¯

i

, or the surface element ds is:

d

⁢

s

=

∥

δ

i

⁢

j

+

Z

i

⁢

Z

j

∥

⁢

d

⁢

x

1

⁢

d

⁢

x

2

⁢

…

⁢

d

⁢

x

n

-

1

The determinant of the matrix H defined by

H

i

⁢

j

=

δ

i

⁢

j

+

Z

i

⁢

Z

j

is the product of its

eigenvalues

. For any (n-1)-vector

W

¯

we have:

H

⁢

W

¯

=

W

¯

+

(

Z

¯

⋅

W

¯

)

⁢

Z

¯

Z

¯

being the (n-1)-vector

(

Z

1

,

Z

2

⁢

…

⁢

Z

n

-

1

)

. If

W

¯

is

orthogonal

to

Z

¯

,

H

⁢

W

¯

=

W

¯

and its eigenvalue is 1. But there are (n-2) such vectors, so the determiant is the last eignevalue for

W

¯

=

Z

¯

:

H

⁢

Z

¯

=

(

1

+

|

Z

|

2

)

⁢

Z

¯

. Finally:

d

⁢

s

=

Z

1

2

+

Z

2

2

+

…

+

Z

n

-

1

2

⁢

d

⁢

x

1

⁢

d

⁢

x

2

⁢

…

⁢

d

⁢

x

n

-

1

References

1

Eric Weinstein -

Hypersphere

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

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

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