{
  "abstract": "We study medial axes of nonempty closed sets in finite-dimensional real normed spaces whose norms are $C^1$ away from the origin. Using the differential of the norm, we define the dual $k$-medial axis by requiring the contact covectors associated with nearest points to have affine dimension at least $k-1$. In dimension $n$, we prove that this set is $(n-k+1)$-rectifiable for every $1\\le k\\le n+1$, and hence has Hausdorff dimension at most $n-k+1$. We give a direct geometric proof and characterize the dual medial axes through the superdifferential of the squared distance function, obtaining an alternative proof from a theorem of Alberti, Ambrosio, and Cannarsa on singularities of semiconvex functions. Under the additional assumption of strict convexity, the ordinary and dual medial axes coincide for $k=2,3$, yielding the corresponding rectifiability results for ordinary medial axes. For $k=4$, however, the ordinary rectifiability conclusion can fail even for norms arbitrarily close to Euclidean: we construct norms on $\\mathbb R^3$ with smooth positively curved unit spheres arbitrarily $C^\\infty$-close to the Euclidean sphere, together with four-point sets whose ordinary $4$-medial axes contain nondegenerate line segments.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30878",
  "author_total": 3,
  "authors": [
    {
      "name": "Arnaud Durand"
    },
    {
      "name": "Xiangyu Liang"
    },
    {
      "name": "Sicheng Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3k5nefjhybgra.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30878",
  "fulltext_url": null,
  "id": "3k5nefjhybgra",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30878",
  "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.CA"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7223377383",
  "pdf_url": "https://arxiv.org/pdf/2610.30878",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3k5nefjhybgra.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30878",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30878"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Dual medial axes in finite-dimensional normed spaces",
  "title_text": "Dual medial axes in finite-dimensional normed spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/3k5nefjhybgra",
  "venue": "arXiv",
  "versions": []
}