{
  "abstract": "We establish sharp weighted Rogers--Shephard inequalities for Rényi differential entropy. For independent copies $X,X'$ of a log-concave random vector in $\\R^n$, we prove $h_p(aX-bX')\\le h_p(X)+n\\log(a+b)$ for all $a,b>0$ and $1\\le p\\le\\infty$, where orders $1$ and $\\infty$ correspond to Shannon entropy and min-entropy, respectively. The Shannon case settles the entropic Rogers--Shephard conjecture of Madiman and Kontoyiannis and its higher-dimensional formulation. For $1\\le p<\\infty$, equality holds precisely for invertible affine images of random vectors with independent exponential coordinates. At $p=\\infty$, the larger equality class is characterized by possibly extended-valued convex potentials that are positively homogeneous of degree one. For every $0<p<1$, uniform laws on sufficiently high-dimensional simplices violate the unweighted bound. Thus $p=1$ is the critical order for this bound to hold in every dimension. The proof combines weighted convex intersections with an exponential lift to a log-affine probability density. At the Shannon endpoint, this lift yields an exact deficit decomposition with a conditional mutual information term for an additive channel with asymmetric Laplace noise.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15716",
  "author_total": 4,
  "authors": [
    {
      "name": "Bang-Xian Han"
    },
    {
      "name": "Guoqing Zhan"
    },
    {
      "name": "Chiyu Zhou"
    },
    {
      "name": "Zhuo-Nan Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/38cxtxqchm8q4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15716",
  "fulltext_url": null,
  "id": "38cxtxqchm8q4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15716",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7221797218",
  "pdf_url": "https://arxiv.org/pdf/2610.15716",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/38cxtxqchm8q4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15716",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15716"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Sharp Rogers--Shephard Inequalities for Rényi Entropy",
  "title_text": "Sharp Rogers--Shephard Inequalities for Rényi Entropy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/38cxtxqchm8q4",
  "venue": "arXiv",
  "versions": []
}