This is 3 related questions, for $L_1, L_2$ and $L_\inf$ norms (in 3d, random splits of an octahedron, sphere, cube).
(Where does this come from ?
In searching for a minimum of a noisy function of $n$ variables,
one may start N parallel searches from N random start points,
e.g. 10 start points in 5d.
Small Voronoi cells are then inefficient, equal-sized cells most efficient.)