{
  "abstract": "Let \\(h\\) be proper, closed and sublinear, and let \\(F\\) be of class \\(C^1\\). The exact \\(\\mathcal{VU}\\) calculus of the composite \\(f=h\\circ F\\) is taken as known. This paper studies enlargements \\(\\bar\\partial_\\varepsilon h\\subset\\partial_\\varepsilon h\\) and the resulting \\(\\varepsilon\\)-spaces of \\(f\\). Nested inclusions give one-sided Painlevé--Kuratowski limits of \\(\\mathcal{V}_\\varepsilon\\) and \\(\\mathcal{U}_\\varepsilon\\) at points of the lineality space \\(\\mathcal{U}h(0)\\). An additional transversality transfers those limits through \\(F\\); without it only the exact chain rule is available. Approximate leaves \\(\\mathcal{M}_\\varepsilon(x)=F^{-1}\\bigl(F(x)+\\mathcal{U}_\\varepsilon h(F(x))\\bigr)\\) are inner semicontinuous under the nesting \\(\\mathcal{U}h(F(\\bar x))\\subset\\mathcal{U}_\\varepsilon h(F(x))\\). For a polyhedral outer function the activity set freezes along the exact leaf for small \\(\\varepsilon\\), so the dimension identity holds on that leaf and not necessarily in a full neighbourhood of \\((\\bar x,0)\\). Partial smoothness of \\(h\\) relative to \\(\\mathcal{U}h(0)\\) extends a known result from the origin to the lineality space; it is not used by the transfer theorems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25083",
  "author_total": 1,
  "authors": [
    {
      "name": "Shuai Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2snzs0nh36md4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25083",
  "fulltext_url": null,
  "id": "2snzs0nh36md4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25083",
  "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.OC"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7223057600",
  "pdf_url": "https://arxiv.org/pdf/2610.25083",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2snzs0nh36md4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25083",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25083"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "$\\varepsilon$-$\\mathcal{VU}$ spaces and approximate leaves\\\\ for sublinear composite functions",
  "title_text": "ε-𝒱𝒰 spaces and approximate leaves\\\\ for sublinear composite functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/2snzs0nh36md4",
  "venue": "arXiv",
  "versions": []
}