Recently I was working through the proof of the Grove-Shiohama maximal diameter sphere theorem, which states:
If $(M,g)$ is a closed Riemannian manifold with sectional curvature bounded below by $1$ and $\mathrm{diam}(M) > \pi/2$ then $M$ is homeomorphic to a sphere.
I was wondering if there was a simple proof of this theorem in the special case when we assume the manifold has constant sectional curvature equal to $1$.
In particular, I don't want to use Toponogov's hinge theorem, and would like to avoid the critical point theory for the distance function introduced by Grove and Shiohama.
Let $G$ be a nontrivial finite isometry group of the unit round sphere $S^n$. Consider an orbit $Gx\subset S^n$ and the associated Voronoi tiling. Each tile $V$, say, $V=V_x$ containing $x$, is contained in a hemisphere centered at $x$ (it equals the intersection of the hemispheres bounded by bisectors of $x, gx$, $g\in G-\{1\}$); hence, the distance from $x$ to each point $y\in V_x$ is at most $\pi/2$. It follows that for the distance between any two points $\bar{x}, \bar{y}\in S^n/G$ is at most $\pi/2$.