{
  "abstract": "A celebrated result in convex geometry is Grünbaum's inequality, which quantifies how much volume of a convex body can be cut off by a hyperplane passing through its barycenter. In this work, we establish a series of sharp Grünbaum-type inequalities -with equality characterizations -for probability measures under certain concavity assumptions. As an application, we apply the renowned Ehrhard inequality and deduce an \"Ehrhard-Grünbaum\" inequality for the Gaussian measure on $R^n$ , which improves upon the bound derived from its log-concavity. For s-concave Radon measures, our framework provides a simpler proof of known results and, more importantly, yields the previously missing equality characterization. This is achieved by gaining new insight into the equality case of their Brunn-Minkowskitype inequality. Moreover, we show that these \"s-Grünbaum\" inequalities can hold only when s > -1. However, for convex measures on the real line, we prove Grünbaum-type inequalities involving their cumulative distribution function.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2507.06759",
  "author_total": 5,
  "authors": [
    {
      "name": "Matthieu Fradelizi"
    },
    {
      "name": "Dylan Langharst"
    },
    {
      "name": "Jiaqian Liu"
    },
    {
      "name": "Francisco Marín Sola"
    },
    {
      "name": "Shengyu Tang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/861v6mcdghnp8.bib",
  "cited_by": 0,
  "doi": "10.1016/j.matpur.2026.103938",
  "fulltext_url": null,
  "id": "861v6mcdghnp8",
  "issns": [],
  "landing_url": "https://hal.science/hal-05777267v1",
  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W4414984957",
  "pdf_url": "https://hal.science/hal-05777267/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/861v6mcdghnp8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:47Z",
      "native_id": "hal-05777267",
      "source": "hal",
      "url": "https://hal.science/hal-05777267v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "Grünbaum's inequality for Gaussian and convex probability measures",
  "title_text": "Grünbaum's inequality for Gaussian and convex probability measures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:47Z",
  "url": "https://research.draconitic.com/paper/861v6mcdghnp8",
  "venue": "HAL",
  "versions": []
}