I would like to pick at random $3$ vectors on the sphere $\mathbb{S}^2$ in $\mathbb{R}^3$ and compute the expected distance of one vector to the span of the others. More precisely: let $a_1$, $a_2$, $a_3 \in \mathbb{S}^2$ be picked uniformly. What can we expect of $\underset{i,j,k\in\{1,2,3\} i\neq j \neq k}{min}dist(a_i, span(a_j,a_k))$? (I am also interested in the higher dimensional analogue)
My approach to this was a bit naïve: two vectors in $\mathbb{S}^2$ define a hyperplane through the origin, I calculate the measure of the set $\{-d \leq e_1 \leq d\}^c\cap \mathbb{S}^2$ and divide it by the total measure of $\mathbb{S}^2$, call it $g$. So given, say $a_1, a_2$, the probability that $a_3$ is closer than $d$ to the span of the other vectors is $1-g$. To calculate the minimum over all 3 combinations, I just use the union bound - so 3(1-g) is the probability the minimum is greater than $d$ (since the respective distances to the span are not independent (?) ). Is there a more refined way to do so (keeping in mind that I would like to use it for bigger dimensions than 3). How to detect the dependencies?
Any suggestion or reference is very much appreciated!
Without loss of generality, let the span of the first two vectors be the $z$ plane. The sphere $\mathbb S^2$ has the special property that the coordinates of uniformly random points on it are uniformly distributed. Thus the expected distance is the expected absolute value of the uniform distribution on $[-1,1]$, which is $\frac12$.