{
  "abstract": "We prove that the normalized star, extreme, and periodic $L_1$-discrepancies with arbitrary nonnegative weights suffer from the curse of dimensionality. A single elementary lemma is used in all three cases: after normalizing the target and every one-dimensional kernel to probability densities, a uniform square-root-affinity gap tensorizes over the coordinates. For every $d\\in \\mathbb{N}$ and $0<\\varepsilon<1$, the normalized information complexities have lower bounds of the form \\[ \\frac{(1-\\varepsilon)^2}{1+\\varepsilon}\\,q^d, \\] with \\[ q_{\\mathrm{star}}=\\frac12+\\frac1{\\sqrt3},\\qquad q_{\\mathrm{ext}}=\\frac{75(9+4\\sqrt2)}{784},\\qquad q_{\\mathrm{per}}=\\frac{25}{16}. \\] All three constants are larger than one. The argument uses only nonnegativity, product integration, the elementary inequalities for the square root, and Cauchy--Schwarz.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.24290",
  "author_total": 1,
  "authors": [
    {
      "name": "Josef Dick"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ay2k4429795y2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.24290",
  "fulltext_url": null,
  "id": "ay2k4429795y2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.24290",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2612|Mathematical Approximation and Integration"
    ]
  ],
  "openalex": "W7171538518",
  "pdf_url": "https://arxiv.org/pdf/2607.24290",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ay2k4429795y2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2607.24290",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.24290"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The curse of dimensionality for geometric $L_1$-discrepancies with nonnegative weights",
  "title_text": "The curse of dimensionality for geometric L_1-discrepancies with nonnegative weights",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/ay2k4429795y2",
  "venue": "arXiv",
  "versions": []
}