The Asymptotic Constant for Optimal Integration of C^k Functions on Bounded Lipschitz Domains: Existence and Dependence on the Volume Only

For a bounded open set D ⊂ R^d let e_n(C^k(D), INT) be the minimal worst-case error of algorithms that use n function values to integrate, over D, the functions all of whose partial derivatives up to order k are bounded by 1. In Oberwolfach Report 6/2020, Novak asked whether the asymptotic constant limsup_n nk/d e_n(C^k(D), INT) depends on D, for bounded Lipschitz domains of volume 1, and whether it is always a limit. We show that for all k, d ≥ 1 there is a constant ck,d ∈ (0, ∞), defined on the unit cube by the derivatives of order k alone, such that lim_n nk/d e_n(C^k(D), INT) = ck,d λ(D)1+k/d for every bounded open set D with the cone condition, in particular for every bounded Lipschitz domain. So the limit exists, at volume 1 it is the same number ck,d for all such D, and the bounds on the derivatives of order less than k do not change it. For every bounded open set, without any regularity, the lower limit is at least ck,d λ(D)1+k/d. For d ≥ 2 a hypothesis on the boundary is needed: there are bounded, connected, Jordan measurable "comb" domains of volume 1 on which nk/d e_n(C^k(D), INT) → ∞. The proof adapts the scheme of Zador and of Graf and Luschgy from quantization theory: self-similarity on the cube, a cutoff argument that identifies the constants with and without boundary conditions, and a boundary layer that is controlled through the cone condition by local polynomial reproduction. For the Sobolev classes L^m_p with 1 < p < ∞ the corresponding asymptotics were proved by Polovinkin (1975, 1989); the case p = ∞ with the full C^k norm treated here is not covered there. The value of ck,d and its dependence on d are not determined. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-17469-002 (E. Novak, Optimal algorithms for some high dimensional problems, Oberwolfach Report 6/2020, item (3)).