{
  "abstract": "We give a new, simpler proof of sharp stability for the Prékopa--Leindler inequality. In contrast to the main arguments from Figalli, van Hintum and Tiba~\\cite{FigalliVanHintumTibaPL}, our proof hinges mostly on the sharp stability theory for the Brunn--Minkowski problem, developed by the same authors \\cite{figalli2024BM}, and on the idea of using a height transportation approach in order to match level sets. As a by-product of our techniques, we are able to prove the sharp dependency on $τ= \\min\\{λ,1-λ\\}$ in the Prékopa--Leindler stability constant whenever one of the inputs is logarithmically concave, producing a new, simplified proof of two of the main results from~\\cite{FigalliVanHintumTibaPL}. Finally, in the same fashion as Kristály and Pîrvuceanu \\cite{kristaly2026quantitative}, we are able to obtain new stability results for the Bakry-Emery log-Sobolev inequality and for the Talagrand inequality. Due to the sharpness of the inputs, these results are optimal in terms of exponents. We moreover show that the $L^1$ nature of the distance in such results \\emph{cannot} in general be improved to any other $L^p$ distance, where $p>1.$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03295",
  "author_total": 2,
  "authors": [
    {
      "name": "João Miguel Machado"
    },
    {
      "name": "João P. G. Ramos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bq5t3hb2mkzkp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03295",
  "fulltext_url": null,
  "id": "bq5t3hb2mkzkp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03295",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03295",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bq5t3hb2mkzkp.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:14Z",
      "native_id": "2611.03295",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03295"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A height transportation approach to the quantitative Prékopa--Leindler inequality",
  "title_text": "A height transportation approach to the quantitative Prékopa--Leindler inequality",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/bq5t3hb2mkzkp",
  "venue": "arXiv",
  "versions": []
}