^{n}as a function of n, it increases up to n=5 and then decreases. The volume is π

^{n/2}/((n/2)!). (By the way, I generally use the factorial notation, not the Γ notation, even when the argument isn't an integer.)

But I hear you complaining that it doesn't make sense to compare volumes in different dimensions! Fair enough. Compare the volume of the unit ball in R

^{n}to the cube circumscribing it, which has volume 2

^{n}. Then the portion of the cube which is inside the ball is f(n) = π

^{n/2}/((n/2)! 2

^{n}). This is

*rapidly*decreasing with

*n*. For n = 2, it's π/4 -- the volume of the unit disc is π, and it can be inscribed in a square of area 4. In n = 3, the unit ball has volume 4π/3 and it's inscribed in a cube of size 8, so we get f(3) = π/6. But f(n) decreases superexponentially. f(10) is about 0.0025, f(20) is about 25 in a billion.

I was surprised that it decayed that quickly -- I'd never bothered to work it out. But if you think about it probabilistically, it kind of makes sense. Namely, a random point in the unit n-cube can be identified with its coordinates (x

_{1}, x

_{2}, ..., x

_{n}). It's in the unit n-sphere if and only if the sum of the squares of those coordinates is less than 1. Let y

_{i}= x

_{i}

^{2}-- then y

_{i}has mean 1/3 and variance 4/45. (That's calculus.) So the sum of the squares of the coordinates is a sum of n such independent random variables, and is thus itself a random variable with mean n/3 and variance 4n/45 -- it's no surprise most of its mass is at n > 1. One could probably use large deviation inequalities to quantify this, but come on, it's 11 at night and I have real work to do.

## 5 comments:

I recently mentioned the pi^{n/2}/(n/2)! result over at John Armstrong's blog, but I didn't mention the curious puzzle this gives rise to.

Take n = 2m, and think of the n-dimensional ball B_n as sitting in C^m, m-dimensional complex space. On the other hand, we have the polydisk

(B_2)^m \subseteq C^m

and the symmetric group on m letters acts on (B_2)^m by permuting coordinates. The orbit space (B_2)^m/S_m has the same volume as B_n.

Is there a continuous volume-preserving map from one of these spaces to the other? (Add on other conditions as you see fit: symplectic, diffeomorphism, etc. -- the question is a little open-ended.)

Have you ever looked at the area and volume of negative dimensional spheres?

There are plenty of other surprises in high-dimensional spaces.

I wrote a post the other day that looks at what happens when you arrange n-spheres in n-dimensional space. I use the ratio of volumes argument you have here to try to understand what is going on.

Todd, the problem/puzzle you mention rang a bell somewhere. I can't remember when or how I came across the following, but does the construction of Blass and Schanuel

http://www.math.lsa.umich.edu/~ablass/geom.html

do the trick?

Yemon, you weren't supposed to give it away! ;-)

Yes, it came from those guys. (I was at the conference where they were first discussing this problem.)

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