{
  "abstract": "We study the relationship between the Cramér transform and Tukey's half-space depth for log-concave probability measures. For the uniform probability measure $μ_K$ on a convex body $K\\subseteq\\mathbb{R}^n$, we prove the sharp pointwise comparison $$Λ_K^*(x)\\leq -\\log q_K(x)\\leq Λ_K^*(x)+\\frac12\\log n+C,\\qquad x\\in\\operatorname{int}(K),$$ where $C$ is an absolute constant. The order $\\log n$ is optimal, as shown by the Euclidean ball. The proof combines exponential tilting, one-dimensional log-concavity, and self-concordance of the Cramér transform. As consequences, we obtain sharp-order moment and tail estimates for $Λ_K^*$ and identify $\\exp(Λ_K^*(x))$, up to polynomial factors in the dimension, with the number of independent samples needed for $x$ to be captured by their random convex hull. We also establish an $O(n^2)$ variance bound for the logarithmic half-space depth and use it to derive a general criterion for sharp thresholds of random convex hulls. In particular, this criterion applies to the uniform measures on $\\ell_p$-balls for every $p>1$. These results establish a quantitative link between large-deviation cost, geometric depth, and sampling complexity in high-dimensional convex geometry.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29972",
  "author_total": 1,
  "authors": [
    {
      "name": "Minas Pafis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1vqhwg5spwgdx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29972",
  "fulltext_url": null,
  "id": "1vqhwg5spwgdx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29972",
  "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.MG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7204908833",
  "pdf_url": "https://arxiv.org/pdf/2608.29972",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1vqhwg5spwgdx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29972",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29972"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Cramér transform, half-space depth and threshold phenomena for convex bodies",
  "title_text": "Cramér transform, half-space depth and threshold phenomena for convex bodies",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/1vqhwg5spwgdx",
  "venue": "arXiv",
  "versions": []
}