Minimal Dispersion on the Sphere

Abstract

Abstract The minimal spherical cap dispersion $${{\,\textrm{disp}\,}}_{\mathcal {C}}(n,d)$$ disp C ( n , d ) is the largest number $$\varepsilon \in (0,1]$$ ε ∈ ( 0 , 1 ] such that, for every n points on the d -dimensional Euclidean unit sphere $$\mathbb {S}^d$$ S d , there exists a spherical cap with normalized area $$\varepsilon $$ ε not containing any of these points. We study the behavior of $${{\,\textrm{disp}\,}}_{\mathcal {C}}(n,d)$$ 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 $${{\,\textrm{disp}\,}}_{\mathcal {C}}(n,d)$$ 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.