We are given a unit origin-centered sphere $S$ in $\mathbb{R}^3$, and three points $\mathbf{x},\mathbf{y},\mathbf{z}\in S$. Let $\mathbf{h}$ be a point selected uniformly at random from $S$ and let $H$ be a hyperplane that passes through the center $\mathbf{0}$ of $S$ and has $\mathbf{h}$ as normal vector. Let $E\_x$ be the event that $H$ separates $\mathbf{x}$ from the other two points $\mathbf{y}$ and $\mathbf{z}$.


**Questions**: What is the probability of $E\_x$? How can we extend this result to express the probability of separating $\mathbf{x}$ from other $n-1$ points when there are $n$ points instead of only three?




---


***Note:*** *I conjecture that the answer to the first question is $\frac12\left(\arccos\left(\mathbf{x}^{\top}\mathbf{y}\right)+\arccos\left(\mathbf{x}^{\top}\mathbf{z}\right)-\arccos\left(\mathbf{y}^{\top}\mathbf{z}\right)\right)$.*



Random planes separating points in $\mathbb{R}^3$
Note that the random vector $G/\|G\|$ is uniformly distributed on the unit sphere in $\mathbb R^n$, where $G$ is a standard Gaussian random vector in $\mathbb R^n$.


So, given an $n$-tuple $(x\_1,\dots,x\_n)$ of unit vectors in $\mathbb R^n$, the probability that the random hyperplane separates $x\_1$ from $x\_2,\dots,x\_n$
is
$$p\_n:=2P(X\_1>0,X\_2<0,\dots,X\_n<0),$$
where $X\_j:=x\_j\cdot G$ for $j=1,\dots,n$, so that the covariance matrix of $(X\_1,\dots,X\_n)$ is the Gram matrix of $(x\_1,\dots,x\_n)$.


Schläfli (1858) obtained a differential recurrence relation for probabilities of the form $p\_n/2$; see e.g. [Plackett](https://www.jstor.org/stable/2332716?seq=1) and references there. Using this recurrence for $n\le4$, $p\_n$ can be expressed as an integral of elementary functions.
As pointed out by [Ruben. p. 213](https://www.jstor.org/stable/2333017?seq=1),
"For dimensionality greater than three (spherical tetrahedra, spherical pentahedra, etc.) the areas can no longer be expressed in terms of elementary functions".


