{
  "abstract": "For a fixed set $U=(u_1,\\ldots,u_m)$ of outer unit normals, we consider the cone-volume map which associates to a vector of support numbers $b\\in\\mathbb R^m_{> 0}$ the cone-volume vector $γ\\in\\mathbb R^m$ of the polytope \\[ P(U,b)=\\{x\\in\\mathbb{R}^n:U^\\top x\\leq b\\}. \\] We study its fibers $S(U,γ)$ over prescribed cone-volume vectors $γ$, with particular emphasis on finiteness and non-uniqueness. We give characterizations of the irreducibility of $U$ in terms of properties of the cone-volume map and its solution sets. In particular, for irreducible $U$, we show that $S(U,γ)$ is finite for all $γ$ outside a set of measure zero in the cone-volume cone, and we give an explicit upper bound for its cardinality. In the planar case, we prove that $S(U,γ)$ has semialgebraic dimension at most one for arbitrary $U$. Moreover, for every $U\\in\\mathcal{U}(2,m)$ with $m\\geq 4$, there exists a cone-volume vector $γ$ for which $S(U,γ)$ contains at least two distinct elements. As a consequence, there exist two distinct smooth planar convex bodies with the same cone-volume measure. Finally, for $U$ in general position, we show that the number of solutions is odd for every cone-volume vector outside the exceptional set.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21533",
  "author_total": 3,
  "authors": [
    {
      "name": "Tom Baumbach"
    },
    {
      "name": "Zhang Chen"
    },
    {
      "name": "Martin Henk"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/880f6kgpw7wt4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21533",
  "fulltext_url": null,
  "id": "880f6kgpw7wt4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21533",
  "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.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7222490109",
  "pdf_url": "https://arxiv.org/pdf/2610.21533",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/880f6kgpw7wt4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21533",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21533"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On the Solution Sets of the Cone-Volume Map",
  "title_text": "On the Solution Sets of the Cone-Volume Map",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/880f6kgpw7wt4",
  "venue": "arXiv",
  "versions": []
}