Random Inscribed Polytopes Have Similar Radius Functions as Poisson–Delaunay Mosaics∗

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 Rn+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 [3], 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. 1998 ACM Subject Classification: I.3.5 Computational Geometry and Object Modeling, G.3 Probability and Statistics, G.2 Discrete Mathematics. 2010 AMS Mathematics Subject Classification: 60D05 Geometric probability and stochastic geometry, 68U05 Computer graphics; computational geometry.