{
  "abstract": "We continue the study of the convex hull $D\\subset\\mathbb{R}^{d+n}$ of a disjunction of $n+1$ polytopes in $\\mathbb{R}^d$, managed by $n$ binary indicator variables. Previously, $D$ had been characterized for arbitrary polytopes when $d\\in\\{1,2\\}$, for polytopes sharing a common constraint matrix (under a technical condition, including hyper-rectangles), and for two generalized cross polytopes ($n=1$). Here we give a complete description of $D$ for arbitrary $n$ and $d$ when all $n+1$ polytopes are generalized cross polytopes. Every vertical facet is described by a normalized support inequality, and such an inequality is facet describing if and only if its equality graph, a bipartite graph on the polytopes and the coordinates, is connected; all vertical facets are generated explicitly from feasible spanning trees. We characterize when such a facet is one of the previously known pairwise inequalities, and we show that pairwise inequalities suffice when $d\\leq3$, but not in general. We establish a bijection between the vertical facets and the maximal cells of a regular subdivision of $Δ_n\\timesΔ_{d-1}$\\,, which yields the sharp bound $n+1+2^d\\binom{n+d-1}{n}$ on the number of facets of $D$, attained for generic data. We also give a polynomial-time separation algorithm that returns a violated facet, which is maximally violated among the normalized support inequalities. Computational experiments show that the facets that are not pairwise inequalities close a large part of the gap left by the pairwise inequalities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05560",
  "author_total": 2,
  "authors": [
    {
      "name": "Yushan Qu"
    },
    {
      "name": "Jon Lee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8ybvgbm7xc4cj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05560",
  "fulltext_url": null,
  "id": "8ybvgbm7xc4cj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05560",
  "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.OC"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7220691783",
  "pdf_url": "https://arxiv.org/pdf/2610.05560",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8ybvgbm7xc4cj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05560",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05560"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "More on Disjunctive Convex Hulls of Generalized Cross Polytopes",
  "title_text": "More on Disjunctive Convex Hulls of Generalized Cross Polytopes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/8ybvgbm7xc4cj",
  "venue": "arXiv",
  "versions": []
}