{
  "abstract": "For every $2<p<\\infty$ and every integer $r\\ge2$, we construct a finite set $T\\subseteq S_{L^p[0,1]}$ such that $|T|\\le2^{Cr^2}$, $γ_2(T)\\le Cr$, and $γ_2(\\conv T)\\ge c r^{3/2-1/p}$. Consequently, for every fixed $p>2$, both estimates in Talagrand's Research Problem~2.11.3 fail in each of the spaces $L^p[0,1]$ and $\\ell_p$. The lower bound follows from a multilevel product principle applied to a scale-separated product of Euclidean simplices. The same construction gives the sharp convexification profile of $\\ell_p^r(\\ell_2^{2^{128r}})$ for $2<p\\le\\infty$ and, by finite representability, quantitative failure in every infinite-dimensional Banach space with cotype index greater than $2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06614",
  "author_total": 2,
  "authors": [
    {
      "name": "Helen W. J. Zhang"
    },
    {
      "name": "Chengdong Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6932sspahmn1f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06614",
  "fulltext_url": null,
  "id": "6932sspahmn1f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06614",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7212002557",
  "pdf_url": "https://arxiv.org/pdf/2609.06614",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6932sspahmn1f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06614",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06614"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Convex-Hull Instability of the $γ_2$-Functional in $L^p$",
  "title_text": "Convex-Hull Instability of the γ_2-Functional in L^p",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/6932sspahmn1f",
  "venue": "arXiv",
  "versions": []
}