# isepiphanic inequality

## Primary tabs

Type of Math Object:
Theorem
Major Section:
Reference
Groups audience:

## Mathematics Subject Classification

### n-dimensonal isoperimetric inequality

This note was suggested by Pahio's entry about the isepiphanic inequality.
The hyper-volume V and hyper-area A of an n-dimensional body satisfy the following inequality:

(A ^ n) / (V ^ (n-1)) >= (n*sqrt(pi))^n / G((n/2) + 1)

G is Euler's gamma function.

For n = 2 or 3, we recover the classical inequalities.

Outline of the proof:

1 - Show that the n-dimensional hyper-ball has the greatest volume for a given area.
2 - Establish the formulas for the volume and area of the n-dimensional hyper-ball.
3 - Take the ratio after proper exponentiation to get a dimensionless constant.

### Re: n-dimensonal isoperimetric inequality

Dear dh2718,
Were it worth to write an entry of this generalisation?
Regards,
Jussi

### Re: n-dimensonal isoperimetric inequality

Dear Jussi, if you think that this note is worth an entry, I'll try to do it. This will take me some time (about one month): I am busy these days, beside, Latex is a headache...

### Re: n-dimensonal isoperimetric inequality

You might also like to add the proof which uses the Bruun Minkowski inequality.