Expected distance between random points on hyperplanes

Question

Let $\mathbf{p}$ be a point on a unit $n$-sphere $S_n$ and let $\mathcal{H}_n$ be the set all hyperplanes passing through the center of $S_n$.

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Question: What is the simplest way to calculate the expected Euclidean distance $\mathbb{E}[d]$ between $\mathbf{p}$ and a hyperplane selected uniformly at random from $\mathcal {H}_n$?

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Should we calculate

$$\mathbb{E}[d]=\mathbb{E}\left[\frac{|x_n|}{\sqrt{\sum_{i=1}^n x_i^2}}\right]$$

where $x_1, x_2, \ldots, x_n$ are the values taken by the i.i.d. random variables $X_1, X_2, \ldots, X_n \sim \mathcal{N}(0,1)$?

Answer 1

Your method would yield the right answer, but it's more complicated than it needs to be. From this answer, to generate a random hyperplane passing through the origin, first pick a random point from the surface of the $n$-sphere. The hyperplane would then be defined by $\langle x,u\rangle=0$.

WLOG, let $\mathbf{p} = (1, \underbrace{0, ..., 0}_{n-1})$. The distance from the hyperplane defined by $\langle x,u\rangle=0$ to $\mathbf{p}$ would then be $\left| \langle u, \mathbf{p}\rangle \right|$. This is then simply $|u_1|$. This is the integral of $|x_1|$ over the $n$-sphere, divided by the "surface area" of the sphere. From this, this is given by $$\frac{\frac{2\pi^{\frac{n-1}{2}}}{\Gamma\left(\frac{n+1}{2}\right)}}{\frac{2\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}\right)}} = \frac{\Gamma\left( \frac{n}{2}-1 \right)}{\Gamma\left(\frac{n-1}{2} \right)\sqrt{\pi}}$$