Minimal dispersion on the sphere

2025/05/16 by Litvak, Alexander E., Sonnleitner, Mathias, Szczepanski, Tomasz
#52B55 #52C17 (Primary) 52A23 #52C45 (Secondary) #FOS: Mathematics #Metric Geometry (math.MG) #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2505.10929

Abstract

The minimal spherical cap dispersion \rm disp_C(n,d) is the largest number ε∈ (0,1] such that, no matter how n points are distributed on the d-dimensional Euclidean unit sphere \mathbbS^d, there is always a spherical cap with normalized area ε not containing any of the points. We study the behavior of \rm 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 \rm 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.

Citations

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Related

• The minimal spherical dispersion
• Random Point Sets on the Sphere --- Hole Radii, Covering, and Separation
• On the Optimality of Random Partial Sphere Coverings in High Dimensions
• A Strong threshold for the size of random caps to cover a sphere
• Spherical cap discrepancy of the Diamond ensemble