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 *L*^{p} (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.

## Posts on high dimensional geometry