Minimal Dispersion on the Sphere

The minimal spherical cap dispersion \,disp\,C(n,d) disp C ( n , d ) is the largest number ε ∈ (0,1] ε ∈ ( 0 , 1 ] such that, for every n points on the d -dimensional Euclidean unit sphere Sᵈ S d , there exists a spherical cap with normalized area ε ε not containing any of these points. We study the behavior of \,disp\,C(n,d) disp C ( n , d ) as n and d grow to infinity. We develop connections to the problems of sphere covering and approximation of the Euclidean unit ball by inscribed polytopes. Existing and new results are presented in a unified way. Upper bounds on \,disp\,C(n,d) disp C ( n , d ) result from choosing the points independently and uniformly at random and possibly adding some well-separated points to close large gaps. Moreover, we study dispersion with respect to intersections of caps.