  Using the geodesic distance on the $n$-dimensional sphere, we study the
expected radius function of the Delaunay mosaic of a random set of points.
Specifically, we consider the partition of the mosaic into intervals of the
radius function and determine the expected number of intervals whose radii are
less than or equal to a given threshold. Assuming the points are not contained
in a hemisphere, the Delaunay mosaic is isomorphic to the boundary complex of
the convex hull in $\mathbb{R}^{n+1}$, so we also get the expected number of
faces of a random inscribed polytope. We find that the expectations are
essentially the same as for the Poisson-Delaunay mosaic in $n$-dimensional
Euclidean space. As proved by Antonelli and collaborators, an orthant section
of the $n$-sphere is isometric to the standard $n$-simplex equipped with the
Fisher information metric. It follows that the latter space has similar
stochastic properties as the $n$-dimensional Euclidean space. Our results are
therefore relevant in information geometry and in population genetics.

Probability;Metric Geometry