As dimension increases, the ratio of volume between a unit ball and a unit cube goes to zero. Said another way, if you have a high-dimensional ball inside a high-dimensional box, nearly all the volume is in the corners. This is a surprising result when you first see it, but it’s well known among people who work with such things.
In terms of Lp (Lebesgue) norms, this says that the ratio of the volume of the 2-norm ball to that of the ∞-norm ball goes to zero. More generally, you could prove, using the volume formula in the previous post, that if p < q, then the ratio of the volume of a p-norm ball to that of a q-norm ball goes to zero as the dimension n goes to infinity.
Proof sketch: Write down the volume ratio, take logs, use the asymptotic series for log gamma, slug it out.
Here’s a plot comparing p = 2 and q = 3.
