{
  "abstract": "We investigate five open problems concerning the Hornich-Hlawka inequality and its generalizations posed by Marinescu and Niculescu. We give constructive answers to three of them. First, we construct an equivalent norm on $\\ell^2$ with no three-dimensional Hornich-Hlawka subspace. Second, we present a Nakano space that is not Hornich-Hlawka. Third, we show that the $α$-power generalization of the Hornich-Hlawka inequality fails to hold on nonzero normed spaces when $α\\in (1, 2) \\cup (2, \\infty)$. This implies that on every nonzero inner product space, the $α$-power generalization of the Hornich-Hlawka inequality holds for all triples precisely when $α\\in (0,1] \\cup \\{2\\}$. For the remaining two problems, we establish reductions. We prove that $\\ell^p(N)$ is Hornich-Hlawka for all $N \\geq 3$ and $2 < p \\leq \\log 3 /(\\log 3 - \\log 2)$ if and only if it is Hornich-Hlawka at $N = 3$ and $p = \\log 3 /(\\log 3 - \\log 2)$. Finally, for $p(\\cdot): (0, 1) \\to [1, 2]$ measurable and integers $k \\geq 0$, we establish a Sobolev-Nakano equivalence: we prove that $W^{k,p(\\cdot)}(0,1)$, equipped with the sum of the Luxemburg norms of its derivatives of orders $0, \\ldots, k$, is Hornich-Hlawka if and only if $L^{p(\\cdot)}(0,1)$ is Hornich-Hlawka.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23785",
  "author_total": 1,
  "authors": [
    {
      "name": "Seokbin Lee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3hgj3pv9wcceq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23785",
  "fulltext_url": null,
  "id": "3hgj3pv9wcceq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23785",
  "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",
      "math.CA"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7222942453",
  "pdf_url": "https://arxiv.org/pdf/2610.23785",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3hgj3pv9wcceq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.23785",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23785"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Constructions and reductions on the Hornich-Hlawka inequality and its generalizations",
  "title_text": "Constructions and reductions on the Hornich-Hlawka inequality and its generalizations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/3hgj3pv9wcceq",
  "venue": "arXiv",
  "versions": []
}