Variant of the Schläfli problem about random points on the hypersphere

Question

I believe it was Ludwig Schläfli who first worked out the probability $m$ points uniformly distributed on the $n$-sphere all lie in the same hemisphere. In the limit of large $n$ this probability switches abruptly from 1 to 0 when $m$ exceeds $2n$. Are there results for other distributions of points on the $n$-sphere? I'm interested in the case where $m/2$ points lie in the all-positive orthant and the other $m/2$ lie in the all-negative orthant.

Answer 1

The probability that, of the $m$ random points on the $n$-sphere (that is, on the unit sphere in $\mathbb R^n$), $m/2$ points lie in the all-positive orthant and the other $m/2$ lie in the all-negative orthant is the same as the probability that, of the $m$ iid standard Gaussian random points, $m/2$ points lie in the all-positive orthant and the other $m/2$ lie in the all-negative orthant, which is $$\sum_{J\in\binom{[m]}{m/2}}P(G_{ij}>0\ \forall i\in J\ \forall j\in[n],\ G_{ij}<0\ \forall i\in[m]\setminus J\ \forall j\in[n]) = \binom m{m/2}2^{-mn},$$ where $[n]:=\{1,\dots,n\}$, $\binom{[m]}{m/2}$ is the set of all subsets of cardinality $m/2$ of the set $[m]$, and the $G_{ij}$'s are iid standard normal random variables (r.v.'s). (Here it is of course assumed that $m$ is even.)

This follows because, if $G_1,\dots,G_n$ are iid standard normal r.v.'s, then the random point $$\frac{(G_1,\dots,G_n)}{\sqrt{G_1^2+\dots+G_n^2}}$$ is uniformly distributed on the $n$-sphere.

Answer 2

For Schläfli's problem, "uniformly distributed on the sphere" is a distraction. You get the same answer for any probability density $\rho$ with the property that $$\rho(\pm x_1\,,\ldots,\,\pm x_m)$$ are all equal, i.e. it only needs to be uniform on each axis through the sphere center. That includes the case above, where the support of $\rho$ has one half of the axis always in the positive orthant. In the limit of large $m$, half of the points will be in the positive orthant when sampled uniformly.