The Covering Radius of Randomly Distributed Points on a Manifold

We derive fundamental asymptotic results for the expected covering radius ρ(XN) for N points that are randomly and independently distributed with respect to surface measure on a sphere as well as on a class of smooth manifolds. For the unit sphere Sd ⊂Rd+1, we obtain the precise asymptotic that Eρ(XN)[N/ log N]1/d has limit [(d+ 1)υd+1/υd] as N →∞, where υd is the volume of the d-dimensional unit ball. This proves a recent conjecture of Brauchart et al. as well as extends a result previously known only for the circle. Likewise, we obtain precise asymptotics for the expected covering radius of N points randomly distributed on a d-dimensional ball, a d-dimensional cube, as well as on a three-dimensional polyhedron (where the points are independently distributed with respect to volume measure). More generally, we deduce upper and lower bounds for the expected covering radius of N points that are randomly and independently distributed on a compact metric measure space, provided the measure satisfies certain regularity assumptions.