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## Ball volume

### Definition

 $p\in \mathbb N$
 $V_p:\mathbb R_+\to \mathbb R_+$ $V_p(r):=\beta^p(B_0(r))$

### Discussion

#### Theorems

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

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

#### Reference

Wikipedia: Volume of an n-ball