WIAS Preprint No. 3192, (2025)

On the Lipschitz continuity of the spherical cap discrepancy around generic point sets


Authors

  • Heitsch, Holger
    ORCID: 0000-0002-2692-4602
  • Henrion, René
    ORCID: 0000-0001-5572-7213

2020 Mathematics Subject Classification

  • 11K38 49K30

Keywords

  • Spherical cap discrepancy, uniform distribution on sphere, Lipschitz continuity, necessary optimality conditions

DOI

10.20347/WIAS.PREPRINT.3192

Abstract

The spherical cap discrepancy is a prominent measure of uniformity for sets on the d-dimensional sphere. It is particularly important for estimating the integration error for certain classes of functions on the sphere. Building on a recently proven explicit formula for the spherical discrepancy, we show as a main result of this paper that this discrepancy is Lipschitz continuous in a neighbourhood of so-called generic point sets (as they are typical outcomes of Monte-Carlo sampling). This property may have some impact (both algorithmically and theoretically for deriving necessary optimality conditions) on optimal quantization, i.e., on finding point sets of fixed size on the sphere having minimum spherical discrepancy.

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