Comment on “Equiprobability, Entropy, Gamma Distributions and Other Geometrical Questions in Multi-Agent Systems”, Entropy 2009, 11, 959-971

Abstract

The volume of the body enclosed by the n-dimensional Lamé curve defined by ${\sum }_{i=1}^n{x}_i^b=E$ is computed.

A recent paper [1] derives asymptotic expressions for the volume of the n-dimensional body defined by $0\le {\sum }_{i=1}^n{x}_i^b\le E$ for $b>0$, $x_i\ge 0$. 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 [2].

Writing $E=R^b$, the volume $V_n\left(R\right)$ is

$$\begin{array}{cc}{\displaystyle V_n\left(R\right)=}\hfill & {\displaystyle \int d{x}_1\cdots \int d{x}_n}\hfill \\ & {}_{0\le {\sum }_{i=1}^n{x}_i^b\le R^b}\hfill \end{array}$$

(1)

By dimensional analysis $V_n\left(R\right)=C_n{R}^n$. Let us now compute the integral

$${\int }_0^{\infty }d{x}_1\cdots {\int }_0^{\infty }d{x}_{n}exp[-(x_1^b+\cdots +x_n^b)]={\left[{\int }_0^{\infty }dxexp[-x^b]\right]}^n={\left[\Gamma \left(1+\frac{1}{b}\right)\right]}^n$$

(2)

by using the change of variables $r={(x_1^b+\cdots +x_n^b)}^{1/b}$ and the volume element $d{V}_n\left(r\right)=n{C}_n{r}^{n-1}dr$ as

$${\int }_0^{\infty }d{V}_n\left(r\right)exp[-r^b]=C_n\Gamma \left(1+\frac{n}{b}\right)$$

(3)

Equaling these two expressions one gets:

$$V_n\left(R\right)=\frac{{\left[\Gamma \left(1+\frac{1}{b}\right)\right]}^n}{\Gamma \left(1+\frac{n}{b}\right)}{R}^n$$

(4)

which is the desired formula. This validates the results in [1], since it coincides with the approximate calculation of that paper in the asymptotic limit $n\to \infty$ although, as proven here, it turns out to be valid for any value of n.