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

Submission Status:in press
  • 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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Metadaten
Author:Holger Heitsch, René Henrion
DOI:https://doi.org/10.2478/udt-2025-0011
Parent Title (English):Unif. Distrib. Theory
Document Type:Article
Language:English
Date of Publication (online):2026/02/04
Date of first Publication:2025/12/31
Release Date:2026/02/04
Tag:Lipschitz continuity; necessary optimality conditions; spherical cap discrepancy; uniform distribution on sphere
Volume:20
Issue:1
First Page:35
Last Page:63
Institutes:Weierstraß-Institut für Angewandte Analysis und Stochastik
Subprojects:B04
Licence (German):License LogoCreative Commons - CC BY - Namensnennung 4.0 International
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