{
  "abstract": "We propose Choquet-type extensions for set-pair functions, an $L^p$-integration of a family of such extensions, and the associated min-max extension constants, and apply these to investigate variational problems and establish sharp bounds for many combinatorial and geometric quantities. Specifically, we use Choquet-type extensions to construct equivalent functional representations of fractional optimization problems with set-variables, which find applications in various settings, such as maxcut and bipartiteness ratio on graph limits, and Cheeger constants on Riemannian manifolds. We further establish the $L^p$ integration of a family of Choquet-type extensions, and apply it to derive spectral bounds for conductance and other combinatorial quantities on measure spaces. A monotonicity inequality on min-max extension constants is proposed, which unifies classical estimates and uncovers new bounds for many geometric and combinatorial quantities, such as torsional rigidity, Cheeger constants, Dirichlet $p$-isoperimetric constant, and $p$-Laplacian eigenvalues, in seemingly unrelated underlying structures----including hypergraphs, graph limits, Riemannian manifolds and metric measure spaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.09606",
  "author_total": 1,
  "authors": [
    {
      "name": "Dong Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d29skw1qt2ne6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.09606",
  "fulltext_url": null,
  "id": "d29skw1qt2ne6",
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  "landing_url": "https://arxiv.org/abs/2608.09606",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
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      "arxiv",
      "math.CO"
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      "math.FA"
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      "2604|Geometric Analysis and Curvature Flows"
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  "pdf_url": "https://arxiv.org/pdf/2608.09606",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d29skw1qt2ne6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2608.09606",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.09606"
    }
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
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    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Choquet-type extensions of set-pair functions, and applications in graph limits, hypergraphs, and metric measure spaces",
  "title_text": "Choquet-type extensions of set-pair functions, and applications in graph limits, hypergraphs, and metric measure spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/d29skw1qt2ne6",
  "venue": "arXiv",
  "versions": []
}