{
  "abstract": "L-extendable functions on the integer lattice are functions that admit a $\\frac12$UJ-convex relaxation; that is, values can be assigned to the additional half-integral points, while keeping the original values at integer points, so that the resulting function is $\\frac12$UJ-convex. Various examples of L-extendable functions and their applications are known. A general characterization of L-extendability for functions on the integer lattice, however, remains open. In this paper, we study the L-extendability of functions on the integer lattice. We first obtain a necessary condition: L-extendable functions are integrally convex. We then consider a special subclass of L-extendable functions, namely, those whose local convex extensions give L-convex relaxations. This subclass has a useful optimization property: the optimal value of the original function coincides with that of the $\\frac12$UJ-convex relaxation given by its local convex extension, a property not guaranteed for general $\\frac12$UJ-convex relaxations. We characterize this subclass by showing that a function is L-extendable through its local convex extension if and only if it is integrally convex and its restriction to every unit hypercube is D-convex, where D-convexity is a discrete convexity associated with reflected Freudenthal simplicial divisions. This subclass strictly contains the class of D-convex functions. We also give a combinatorial characterization of D-convex subsets of the vertices of a unit hypercube: a set $X\\subseteq\\{0,1\\}^n$ is D-convex if and only if it is majority-closed and contains a relative opposite pair.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38287",
  "author_total": 3,
  "authors": [
    {
      "name": "Ken Yokoyama"
    },
    {
      "name": "Yuni Iwamasa"
    },
    {
      "name": "Kei Kimura"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9tt24njxn4g98.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38287",
  "fulltext_url": null,
  "id": "9tt24njxn4g98",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38287",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38287",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9tt24njxn4g98.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38287",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38287"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Characterization of Functions on the Integer Lattice That Are L-Extendable through Local Convex Extensions",
  "title_text": "A Characterization of Functions on the Integer Lattice That Are L-Extendable through Local Convex Extensions",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/9tt24njxn4g98",
  "venue": "arXiv",
  "versions": []
}