Let $\mathbb{S}^n\subseteq\mathbb{R}^{n+1}$ be the sphere with geodesic distance, and let $p_1,\dots,p_{n+2}\in\mathbb{S}^n$ be distinct points forming a regular simplex in $\mathbb{R}^{n+2}$. For $i=1,\dots,n+2$, let $V_i:=\{x\in\mathbb{S}^n;d(x,p_i)<d(x,p_j)\text{ for all }j\neq i\}$ be the Voronoi cell associated to $p_i$.
What is the diameter of the sets $V_i$? I remember I read it somewhere but I don't know where.