# A nice formula for the volume of an L_p ball

I recently came across this paper:

Volumes of Generalized Unit Balls
Xianfu Wang
Mathematics Magazine, Vol. 78, No. 5 (Dec., 2005), pp. 390-395

which has nice formula for a “generalized unit ball” in $\mathbb{R}^n$:

$\mathbb{B}_{p_1,p_2,\ldots,p_n} = \{ \mathbf{x} = (x_1, x_2, \ldots, x_n) : |x_1|^{p_1} + |x_2|^{p_2} + \cdots + |x_n|^{p_n} \le 1 \}$

These balls can look pretty crazy (as some pictures in the paper show).

The main result is that for $p_1, \ldots, p_n > 0$, the volume is equal to

$\mathrm{Vol}(\mathbb{B}_{p_1,\ldots,p_n}) = 2^n \frac{ \Gamma(1 + 1/p_1) \cdots \Gamma(1 + 1/p_n) }{ \Gamma(1/p_1 + 1/p_2 + \cdots + 1/p_n + 1) }$

The formula for the volume of the $n$-sphere in the $L_2$ norm is well known, but this formula lets us calculate all sorts of volumes. For example, for the unit $L_1$ ball we get the rather clean and beautiful formula

$2^n \frac{\Gamma(2)^n}{\Gamma(n + 1)} = \frac{2^n}{n!}$

The proof given in the note is by induction, and a remark at the end points to several other proofs based on Laplace transforms.