Intuitive proof that the first (n-2) coordinates on a sphere are uniform in a ball - MathOverflow

Intuitive proof that the first (n-2) coordinates on a sphere are uniform in a ball

2010-07-23T19:29:10Z

It is a classical fact that if $(x_1,\ldots,x_n)$ is a random vector uniformly distributed on the sphere $S^{n-1} \subseteq \mathbb{R}^n$, then the random vector $(x_1,\ldots,x_{n-2})$ is uniformly distributed in the unit ball $B_{n-2} = \{ (y_1,\ldots,y_{n-2}) \mid \sum_{i=1}^{n-2} y_i^2 \le 1\} \subseteq \mathbb{R}^{n-2}$ . In measure-theoretic language, the pull-back of volume measure on $B_{n-2}$ via the coordinate projection $S^{n-1} \to B_{n-2}$, $(x_1,\ldots,x_n) \mapsto (x_1,\ldots,x_{n-2})$ is Hausdorff measure on $S^{n-1}$ (up to normalization). Apparently the $n=3$ case was known to Archimedes.

Is there an intuitive geometric proof of this, that in particular explains why you drop 2 coordinates, as opposed to 1 or 3 or ...? Or even some heuristic that explains the 2?

I already know reasonably slick probabilistic proofs of this result, including a version for $\ell_p$ norms when $p$ is an integer and you project onto the first $n-p$ coordinates (using the right distribution on the $\ell_p$ sphere, which is not surface area except for $p=1,2$), but as far as I can see they just make it look like a coincidence that things turn out this way. (And as far as I know, maybe it is.)

Answer by Greg Kuperberg for Intuitive proof that the first (n-2) coordinates on a sphere are uniform in a ball

2010-07-23T20:05:56Z

One viewpoint, which is a bit gauche for your construction but valid, is that the general result is a corollary of Archimedes' theorem, that the projection from a 2-sphere $S^2$ to an interval $I$ is measure-preserving. Whether or not you view it as a coincidence, Archimedes' theorem has an important generalization. Namely, $S^2$ is the simplest example of a projective toric variety, and the coordinate projection is its toric moment map. The moment map of any projective toric variety is measure preserving. For instance, the moment map from $\mathbb{C}P^n$ to the $n$-simplex shows you that the Fubini-Study volume of the former is $\pi^n/n!$. You might also recognize this as the volume of the unit ball $B_{2n}$. There is a simple symplectic map from $B_{2n}$ to $\mathbb{C}P^n$ which is 1-to-1 in the interior and quotients the boundary to $\mathbb{C}P^{n-1}$. (I learned/realized these facts in an old discussion with Doug Ravenel and Yael Karshon.)

So you could say that the original relation has a good explanation in complex and symplectic geometry, and that the explanation has been disguised a bit in real geometry. Moreover, that 2 arises because $\dim_\mathbb{R} \mathbb{C} = 2$.

Answer by Dick Palais for Intuitive proof that the first (n-2) coordinates on a sphere are uniform in a ball

2010-07-25T10:29:11Z

I find myself very confused by all this, and I suspect I must be missing something very important, and I am hoping someone (Greg?) can set me straight.

Let's just consider the classic case n=3, so we are projecting from the 2-sphere S onto its projection on the $x$-axis, i.e., the interval $I = [-1,1]$ using the map $(x,y,z) \mapsto x$.

The Archimedes projection that has been mentioned several times is very different---it is the projection of S to the right circular cylinder C tangent to the sphere along its equator. I agree that this is measure (i.e., area) preserving. (Who am I to argue with Archimedes?) On the other hand, the projection mentioned by the OP is dimension reducing so we seem to be comparing the area of a region with length of its projection.

Now the projection of S onto I can't be measure preserving can it? First of all it doesn't seem to be dimensionally correct. Consider for example a small spherical cap of radius $r$ centered at $(0,0,1)$. To first order its area is $\pi r^2$. However, its projection is the interval $[-r,r]$ which has length $2r$. How can these quantities be proportional? Even worse, if we apply a rotation, the area of the spherical cap stays constant, but the area of its projection varies wildly.

It sounds to me like I must be somehow misinterpreting the original question, but I haven't been able to re-interpret it in a way consistent with its wording.