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Journal

Uniform Distribution Theory (UDT)


Volume & Issue

UDT. Volume 20. Issue 1 & 2


Published on

December, 2025


Pages

224-254


DOI

http://doi.org/10.2478/UDT-2025-0011

Article

On the Lipschitz Continuity of the Spherical Cap Discrepancy around Generic Point Sets


Authors

Holger Heitsch and René Henrion


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.


Citation

Heitsch, H. & Henrion, R. (2025). On the Lipschitz Continuity of the Spherical Cap Discrepancy around Generic Point Sets. Uniform distribution theory, 20(1), 224-254. https://doi.org/10.2478/udt-2025-0011