{
  "abstract": "We establish dimension-free, pointwise optimal lower-deviation bounds for convex functions under spherically symmetric Cauchy-type measures. These bounds use either mean-variance normalization or moments about the median, the latter covering the Cauchy case where the mean and variance are unavailable. By approximation with a parametrized family the Cauchy-type measures, our results recover the Gaussian setting of Paouris and Valettas \\cite{PaourisValettas18} and strengthen their variance-sensitive estimate to a sharp bound, with a complete equality characterization. This goes beyond the partial improvements obtained in subsequent work. We also prove an Ehrhard-type inequality for arbitrary convex sets under these Cauchy-type measures in every dimension and characterize all equality cases. This resolves a conjecture posed by Byczkowski, Małecki and Żak \\cite{ByczkowskiMaleckiZak22}. The proof follows from a variational argument combined with the joint concavity of fiber rearrangement. The geometric inequality reduces distributions of convex observables to one-dimensional convex transforms, yielding the sharp lower-deviation bounds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14880",
  "author_total": 3,
  "authors": [
    {
      "name": "Yong Huang"
    },
    {
      "name": "Shengyu Tang"
    },
    {
      "name": "Sinan Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fbh05kckp4sds.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14880",
  "fulltext_url": null,
  "id": "fbh05kckp4sds",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14880",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7221770379",
  "pdf_url": "https://arxiv.org/pdf/2610.14880",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fbh05kckp4sds.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14880",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14880"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "From Cauchy to Gaussian: Sharp Lower Deviations and Ehrhard-Type Inequality",
  "title_text": "From Cauchy to Gaussian: Sharp Lower Deviations and Ehrhard-Type Inequality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/fbh05kckp4sds",
  "venue": "arXiv",
  "versions": []
}