{
  "abstract": "Let $X$ be a normed space and $S_X$ be the unit sphere of $X$. We say that $X$ has Property (H) if there exist two increasing sequences $V_j\\subset X$ and $H_j\\subset \\ell_2$, $j=1,2,\\ldots$, of finite dimensional subspaces such that $V=\\bigcup_j V_j$ is dense in $X$, and there is a uniformly continuous mapping $ψ:S_V\\to S_{\\ell_2}$ such that the restrictions $ψ|_{S_{V_j}}$, $j=1,2,\\ldots$, are isomorphisms from $S_{V_j}$ onto $S_{H_j}$. The Kasparov-Yu problem asks whether the Banach space $c_0$ of all null sequences admits Property (H), or rational Property (H). In this paper, we prove that $c_0$ admits neither Property (H) nor rational Property (H). We also obtain the following results. (1) The infimum $μ_n$ of the Lipschitz constants of all nonzero-degree maps from $S_{\\ell_\\infty^n}$ to $S_{\\ell_2^n}$ satisfies $\\lim_{n\\to\\infty} μ_n/\\log n=1/2$. (2) Let $λ_n$ and $C_n$ denote the infima of the Lipschitz constants of homeomorphisms from $S_{\\ell_\\infty^n}$ onto $S_{\\ell_2^n}$ and from $B_{\\ell_\\infty^n}$ onto $B_{\\ell_2^n}$, respectively. Then $\\lim_{n\\to\\infty} λ_n/\\log n=\\lim_{n\\to\\infty} C_n/\\log n=1/2$. The first result solves a problem of Gromov on nonzero-degree maps between finite-dimensional spheres, while the second settles Johnson's problem on the Lipschitz distortion of finite-dimensional unit balls.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15817",
  "author_total": 3,
  "authors": [
    {
      "name": "Lixin Cheng"
    },
    {
      "name": "Qingjin Cheng"
    },
    {
      "name": "Yue Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9anwt0p0v35hn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15817",
  "fulltext_url": null,
  "id": "9anwt0p0v35hn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15817",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7213249197",
  "pdf_url": "https://arxiv.org/pdf/2609.15817",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9anwt0p0v35hn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.15817",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15817"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On Property (H) of c_0 and Lipschitz Problems of Gromov and Johnson",
  "title_text": "On Property (H) of c_0 and Lipschitz Problems of Gromov and Johnson",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/9anwt0p0v35hn",
  "venue": "arXiv",
  "versions": []
}