Ratio of Lebesgue norm ball volumes

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 a-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.

• Corners stick out more in high dimensions
• The cross polytope