{
  "abstract": "We establish the Alexandrov-Fenchel inequality and its modified version for arbitrary, possibly nonsmooth, horospherically convex bodies in hyperbolic space $\\mathbb{H}^{n+1}, n\\geq 2$, and prove that equality in each inequality holds if and only if the h-convex body is a geodesic ball. A key ingredient is a rigidity theorem characterizing geodesic balls via proportionality between modified support measures of adjacent orders. Its proof combines the $C^{1,α}\\cap W^{2,k+1}$ approximation of the outer parallel hypersurfaces by smooth strictly h-convex hypersurfaces with the estimates for the total variation norm of the signed measure associated with the proportionality relation along the shifted curvature flows independent of initial curvature bounds. For $0\\leq k\\leq n-1$, we construct weak flows preserving the $k$-th modified quermassintegral $\\tilde{W}_k$ with arbitrary h-convex initial bodies as Hausdorff limits of the smooth flows. The rigidity theorem above characterizes the limit shape of the flow and gives a positive lower bound on the rate of decrease of $\\tilde{W}_{k+1}$ for a short time whenever the initial h-convex body is not a geodesic ball, which characterizes the equality cases in the modified Alexandrov-Fenchel inequality. For the Alexandrov-Fenchel inequality, we establish the Hausdorff stability independent of curvature bounds described by the Alexandrov-Fenchel deficits for h-convex bodies, which characterizes the equality cases in the Alexandrov-Fenchel inequality.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18761",
  "author_total": 1,
  "authors": [
    {
      "name": "Xinxiu Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6qe5dyx07b7jn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18761",
  "fulltext_url": null,
  "id": "6qe5dyx07b7jn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18761",
  "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.DG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7221778036",
  "pdf_url": "https://arxiv.org/pdf/2610.18761",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6qe5dyx07b7jn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.18761",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18761"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Modified support measures and Alexandrov-Fenchel type inequalities for nonsmooth h-convex bodies in hyperbolic space",
  "title_text": "Modified support measures and Alexandrov-Fenchel type inequalities for nonsmooth h-convex bodies in hyperbolic space",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/6qe5dyx07b7jn",
  "venue": "arXiv",
  "versions": []
}