Average chord length on a unit $(n-1)$-dimensional sphere in $n$-dimensional space

Question

How to find the average chord length on a sphere of radius $1$? It is clear that this cannot be done as long as there is no law of distribution of its endpoints. Two probability distributions seem logical:

1. Two points uniformly distributed on the sphere are taken and connected by a segment;
2. A random direction is taken and parallel to it, uniformly along the projection of the sphere onto a plane perpendicular to this direction, a segment inside the ball is taken.

It turns out that these are completely different models!

In model 1), the average chord length for all $n$ is greater than $\frac{1}{2}$, since the second point with probability $\frac{1}{2}$ is in another hemisphere and the distance in this case is greater than $1$. In model 2), the average chord length tends to zero! Since it is equal to the volume of the $n$-dimensional ball divided by the volume of the $(n-1)$-dimensional ball.

Such nonsense happens in the $n$-dimensional case, I want to figure it out! And how to calculate exactly the average chord length in 1) case?

Answer 1

We can find the average in case 1) by using polar coordinates in $n$ dimensions. As endpoints of the chord we can take $$ \begin{align} &P=(1,0,\dots,0),\\ &Q=(\cos\phi_1, \sin\phi_1\cos\phi_2,\dots, \sin\phi_1\cdots\sin\phi_{n-2}\cos\phi_{n-1}, \sin\phi_1\cdots\sin\phi_{n-2}\sin\phi_{n-1}), \end{align} $$ so that: $$ PQ^2=(1-\cos\phi_1)^2+\sin^2\phi_1=2(1-\cos\phi_1). $$ We can then integrate that over the whole sphere to get the average value of $PQ$: $$ \langle PQ\rangle={ \int_0^\pi\int_0^\pi\cdots\int_0^{2\pi} \sqrt{2(1-\cos\phi_1)} \sin^{n-2}\phi_1\sin^{n-3}\phi_2\cdots \sin\phi_{n-2}\,d\phi_1\,d\phi_2\cdots d\phi_{n-1} \over \int_0^\pi\int_0^\pi\cdots\int_0^{2\pi} \sin^{n-2}\phi_1\sin^{n-3}\phi_2\cdots \sin\phi_{n-2}\,d\phi_1\,d\phi_2\cdots d\phi_{n-1} }. $$ Note that all integrals factor out, with the exception of that in $\phi_1$: $$ \langle PQ\rangle={ \int_0^\pi \sqrt{2(1-\cos\phi_1)} \sin^{n-2}\phi_1\,d\phi_1 \over \int_0^\pi \sin^{n-2}\phi_1\,d\phi_1 } =\frac{2^{n-1} \left(\Gamma \left(\frac{n}{2}\right)\right)^2} {\sqrt{\pi }\,\Gamma\left(n-\frac{1}{2}\right)}. $$