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We derive fundamental asymptotic results for the expected covering radius   $\rho (X_N)$   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   $\mathbb {S}^d \subset \mathbb {R}^{d+1}$  , we obtain the precise asymptotic that   $\mathbb {E}\rho (X_N)[N/\log N]^{1/d}$   has limit   $[(d+1)\upsilon _{d+1}/\upsilon _d]^{1/d}$   as   $N \to \infty $  , where   $\upsilon _d$   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.

The Covering Radius of Randomly Distributed Points on a Manifold