{
  "schema_version": 1,
  "problem_number": "OWR-17469-002",
  "title": "The Asymptotic Constant for Optimal Integration of C^k Functions on Bounded Lipschitz Domains: Existence and Dependence on the Volume Only",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "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 n^{k/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 c_{k,d} ∈ (0, ∞), defined on the unit cube by the derivatives of order k alone, such that lim_n n^{k/d} e_n(C^k(D), INT) = c_{k,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 c_{k,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 c_{k,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 n^{k/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 c_{k,d} and its dependence on d are not determined. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.NA",
    "math.CA",
    "math.FA"
  ],
  "keywords": [
    "numerical integration",
    "cubature",
    "minimal error",
    "asymptotic constant",
    "information-based complexity",
    "functions with bounded partial derivatives",
    "Lipschitz domains",
    "cone condition",
    "quantization",
    "Zador",
    "Polovinkin",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-17469-002",
    "math.NA",
    "math.CA",
    "math.FA",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-20",
  "publication_date": "2026-10-20",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-20",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-17469-002/",
  "pdf_url": "https://eulersolve.org/papers/owr-17469-002/paper.pdf?v=2c9d1a598f45",
  "doi": "10.5281/zenodo.23486812",
  "zenodo_record_url": "https://zenodo.org/records/23486812",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Proved for bounded open sets with the cone condition, in particular all bounded Lipschitz domains; a boundary hypothesis is needed for d >= 2.",
  "files": {
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    "source.zip": {
      "sha256": "e175ad56582b9d6daa5a62cb850f8a507294e2c470fe73007a6ef18d9d5e837f"
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    "verification_report.md": {
      "sha256": "01599d28857be4dc72b7b17f7a70dec3a49f16920a34892cf7ca6017cdeaa3dd"
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}