Spherical cap L2 discrepancy – Blessing of dimensionality and a distance-kernel variant

We prove a dimension-explicit upper bound for the information-complexity (also called the inverse of the discrepancy) of the classical spherical cap 𝐿 2 discrepancy on the d -dimensional sphere 𝕊 𝑑 . For every fixed 𝜀 ∈ ( 0 , 1 ) , this bound implies that a single point suffices in all sufficiently large dimensions, which shows a “blessing of dimensionality” for the associated numerical integration problem. We then introduce a one-parameter family of distance reproducing kernels for the Sobolev space 𝐻 ( 𝑑 + 1 ) / 2 ⁡ ( 𝕊 𝑑 ) . For the balanced kernel 𝐾 ⁡ ( 𝒙 , 𝒚 ) = 1 − 1 √ 2 ⁢ ‖ 𝒙 − 𝒚 ‖ , a lower bound together with a random-point upper bound shows that the normalized information-complexity is Θ 𝜀 ⁡ ( 𝑑 ) for every fixed 𝜀 ∈ ( 0 , 1 ) . Thus its dependence on d is determined up to constants depending only on ε . Finally, we study the one-sided spherical cap discrepancy obtained by restricting the height parameter to [ 0 , 1 ] . We derive its kernel and Stolarsky's invariance principle explicitly. A sharp mean-distance inequality, attained by regular simplices, determines its normalized information-complexity exactly in all sufficiently large dimensions and proves strong polynomial tractability with optimal ε -exponent two.