A set of $8$ points in an unit cube is given, such that the distance between any two points belonging to the set is not less than $1$. Find the distance between the center of the cube and the closest to it point from the set.
Intuitively, I think, each of the points in the set is a vertex for the cube and the distance is $\frac{\sqrt3}{2}$, but I don't know how to prove it.