Given $p\in\mathbb{R^+}$ and $n\in\mathbb{N}$, I am looking to determine a function, $V(p,n)\in\mathbb{R^+}$, which returns the n-dimensional euclidean volume enclosed by the collection of points: $$S = \{z :\sum_{i=1}^{n} |z_i|^p = 1 \}, \quad z \in \mathbb{R}^n.$$
For $n=2$, the function equals: $$V(p, 2) = \frac{ 4\Gamma(1+\frac1p)^2}{\Gamma(1+\frac2p)}.$$
But I do not know a reliable way to determine this for higher integer $n$.
Despite what the title says, it looks like it's the volume of an n-dimensional supercircle that you want. This wiki page gives the formula: $$V(p,n) = \frac{(2\Gamma(\frac{1}{p}+1))^n}{\Gamma(\frac{n}{p}+1)}.$$
This in turn comes from:
Dirichlet, P. G. Lejeune (1839). "Sur une nouvelle méthode pour la détermination des intégrales multiples" [On a novel method for determining multiple integrals], Journal de Mathématiques Pures et Appliquées. 4: 164–168.