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# isepiphanic inequality

The classical *isepiphanic inequality*

$36\pi V^{2}\;\leqq\;A^{3},$ |

concerns the volume $V$ and the area $A$ of any solid in $\mathbb{R}^{3}$. It asserts that the ball has the greatest volume among the solids having a given area.

For a ball with radius $r$, we have

$V\;=\;\frac{4}{3}\pi r^{3},\qquad A\;=\;4\pi r^{2},$ |

whence it follows the equality

$36\pi V^{2}\;=\;A^{3}.$ |

Cf. the isoperimetric inequality and the isoperimetric problem.

# References

- 1 Patrik Nordbeck: Isoperimetriska problemet eller Varför ser man så få fyrkantiga träd? Examensarbete. Lund University (1995).

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## Mathematics Subject Classification

51M25*no label found*51M16

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## Comments

## 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.