Ball volume

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$ p\in \mathbb N $

$V_p:\mathbb R_+\to \mathbb R_+$

$V_p(r):=\beta^p(B_0(r))$

For all $a\in \mathbb R^p$, the volume of the ball $B_a(r)$ is the same and given by

$V_p(r)= \pi^{p/2}\ \Gamma(p/2+1)^{-1}\ r^p = \frac{\pi}{2}^{p/2}\ \Pi(p/2) r^p $