The unit ball in n dimensions under the Lp norm has volume
I ran across this formula via A nice formula for the volume of an L_p ball. That post gives an even more general result that allows different values of p along each axis.
Note that as p goes to ∞ the volume goes to 2n because the terms involving gamma functions go to 1. This is as we’d expect since the unit “ball” in the infinity norm is a cube, two units wide on each side.
Related post: Means and inequalities