I want to select M points on the N-sphere such that $min\_{i\neq j,i,j\in \{1..M\}} ||x\_i - x\_j||$ is maximized.


Are there good upper bounds for this max-min distance?



Bound on maximum distance between points on a unit N-Sphere
Call the quantity in question $D(M, N),$ and let the volume of the spherical cap of dimension $N$ and radius $r$ $V(N, r),$normalized so that the volume of the whole sphere is $1.$ Since the caps of radius $2\arcsin D(M, N)/4$ are disjoint, you know that:


$$
M V(N, 2\arcsin D(M, N)/4) < 1.
$$
Since The function $V(N, r)$ is [well understood](http://en.wikipedia.org/wiki/Spherical_cap), this gives you a bound.


