# Random Inscribed Polytopes Have Similar Radius Functions as Poisson-Delaunay Mosaics

## TL;DR

This paper analyzes the expected radius function of Delaunay mosaics on the sphere, showing that random inscribed polytopes exhibit similar properties to Poisson-Delaunay mosaics in Euclidean space, with implications for information geometry.

## Contribution

It establishes that the expected radius functions of random inscribed polytopes on the sphere are comparable to those of Poisson-Delaunay mosaics in Euclidean space, linking stochastic geometry with information geometry.

## Key findings

- Expected number of small-radius intervals matches Euclidean case
- Inscribed polytopes have similar face count expectations as Poisson-Delaunay mosaics
- Results are relevant for applications in information geometry and population genetics

## Abstract

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.