Diameters of spherical Alexandrov spaces and constant curvature one orbifolds

Let ${Q\\sp{n}} = S\\sp{n}/G,$ where $G\\subset O(n + 1)$ is closed and non-transitive. If G is finite then $Q\\sp{n}$ is an orbifold of constant curvature one. Otherwise, $Q\\sp{n}$ is a spherical Alexandrov space. I will discuss conditions for the diameter of $Q\\sp{n}$ to be $\\pi/2$ or $\\pi,$ and my ongoing research to find lower bounds for these diameters. I will show that there is an explicit lower bound on the diameter when the group is finite, and I will show existence of a lower bound when the group is infinite. In the process, I will compute some resulting quotient spaces and their diameters, such as orbifold lens spaces, the Impossiball (a spherical Rubik's cube which has a subgroup of the icosahedral group acting on it) and Coxeter orbifolds.