{
  "abstract": "In this paper we construct new examples of diameter graphs and Reuleaux polyhedra in $\\mathbb{R}^3$, obtaining a full characterization of their combinatorial structure. For a finite set of points $X\\subset\\mathbb{R}^d$, its diameter graph is the graph on vertex set $X$ where pairs forming a diameter pair are connected by an edge. Grünbaum, Heppes and Straszewicz independently proved that the diameter graph of $X\\subset \\mathbb{R}^3$ has at most $2|X|-2$ edges, answering a question of Vázsonyi. Their proof relied on ball polytopes. The ball polytope $\\mathcal{B}(X)$ is the intersection of the unit balls centered at the points of $X$. We call a ball polytope a Reuleaux polyhedron if the centers form a family with $2|X|-2$ diameter pairs. Kupitz, Martini and Perles showed that the skeleton of a Reuleaux polyhedron must be a 2-connected strongly involutive self-dual graph. They conjectured that in the simple 3-connected case this is also sufficient. We not only confirm this conjecture, but we show that any 2-connected (not necessarily simple) strongly involutive self-dual graph arises as the skeleton of a Reuleaux polyhedron. To construct the new Reuleaux polyhedra we construct new diameter graphs. It was known that any 3-dimensional diameter graph is a subgraph of a non-bipartite quadrangulation of the projective plane. We show that the reverse holds. That is, for any such graph we construct a diameter realization. This also confirms and strengthens a conjecture of Montejano, Pauli, Raggi, Roldán-Pensado on metric embeddings of strongly involutive self-dual graphs. The construction relies on ideas from rigidity theory. We also discuss a number of applications of these results, such as the construction of bodies of constant width and connections to Steinitz's theorem and Borsuk's conjecture.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06340",
  "author_total": 1,
  "authors": [
    {
      "name": "Gábor Damásdi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dta50red70rpn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06340",
  "fulltext_url": null,
  "id": "dta50red70rpn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06340",
  "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.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7220706203",
  "pdf_url": "https://arxiv.org/pdf/2610.06340",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dta50red70rpn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06340",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06340"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Geometric Realizations with Strong Self-Duality Part II: Diameter Graphs, Reuleaux Polyhedra, and Thrackles",
  "title_text": "Geometric Realizations with Strong Self-Duality Part II: Diameter Graphs, Reuleaux Polyhedra, and Thrackles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/dta50red70rpn",
  "venue": "arXiv",
  "versions": []
}