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isepiphanic inequality

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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.

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

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...

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

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