296 2019 eng article 0 2019-12-17 -- -- The spherical cap discrepancy is a widely used measure for how uniformly a sample of points on the sphere is distributed. Being hard to compute, this discrepancy measure is typically replaced by some lower or upper estimates when designing optimal sampling schemes for the uniform distribution on the sphere. In this paper, we provide a fully explicit, easy to implement enumerative formula for the spherical cap discrepancy. Not surprisingly, this formula is of combinatorial nature and, thus, its application is limited to spheres of small dimension and moderate sample sizes. Nonetheless, it may serve as a useful calibrating tool for testing the efficiency of sampling schemes and its explicit character might be useful also to establish necessary optimality conditions when minimizing the discrepancy with respect to a sample of given size. 10.1016/j.cam.2021.113409 in press 2 Creative Commons - CC BY-NC-ND - Namensnennung - Nicht kommerziell - Keine Bearbeitungen 4.0 International Holger Heitsch René Henrion eng uncontrolled spherical cap discrepancy eng uncontrolled uniform distribution on sphere eng uncontrolled optimality conditions Weierstraß-Institut für Angewandte Analysis und Stochastik B04 https://opus4.kobv.de/opus4-trr154/files/296/CapDiscrepancy19.pdf