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Entropy 2009, 11(4), 1121-1122; https://doi.org/10.3390/e11041121

Discussion
Comment on “Equiprobability, Entropy, Gamma Distributions and Other Geometrical Questions in Multi-Agent Systems”, Entropy 2009, 11, 959-971
IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos, CSIC-UIB, Campus UIB, E-07122 Palma de Mallorca, Spain
Received: 11 December 2009 / Accepted: 18 December 2009 / Published: 22 December 2009

## Abstract

:
The volume of the body enclosed by the n-dimensional Lamé curve defined by $∑ i = 1 n x i b = E$ is computed.
Keywords:
Lamé curves
A recent paper  derives asymptotic expressions for the volume of the n-dimensional body defined by $0 ≤ ∑ i = 1 n x i b ≤ E$ for $b > 0$, $x i ≥ 0$. This is the body enclosed by a Lamé curve in n dimensions. Here I compute exactly this volume by using a straightforward modification of the calculation that gives the volume of the n-dimensional sphere, the case $b = 2$, see .
Writing $E = R b$, the volume $V n ( R )$ is
$V n ( R ) = ∫ d x 1 ⋯ ∫ d x n 0 ≤ ∑ i = 1 n x i b ≤ R b$
By dimensional analysis $V n ( R ) = C n R n$. Let us now compute the integral
$∫ 0 ∞ d x 1 ⋯ ∫ 0 ∞ d x n exp [ - ( x 1 b + ⋯ + x n b ) ] = ∫ 0 ∞ d x exp [ - x b ] n = Γ 1 + 1 b n$
by using the change of variables $r = ( x 1 b + ⋯ + x n b ) 1 / b$ and the volume element $d V n ( r ) = n C n r n - 1 d r$ as
$∫ 0 ∞ d V n ( r ) exp [ - r b ] = C n Γ 1 + n b$
Equaling these two expressions one gets:
$V n ( R ) = Γ 1 + 1 b n Γ 1 + n b R n$
which is the desired formula. This validates the results in , since it coincides with the approximate calculation of that paper in the asymptotic limit $n → ∞$ although, as proven here, it turns out to be valid for any value of n.

## Acknowledgements

I acknowledge financial support by the MEC (Spain) and FEDER (EU) through project FIS2007-60327.

## References

1. López-Ruiz, R.; Sañudo, J.; Calbet, X. Entropy 2009, 11, 959–971.
2. Pathria, R.K. Appendix C. In Statistical Mechanics, 2nd ed.; Butterwort-Heinemann: Stoneham, MA, USA, 1996. [Google Scholar]