{
  "abstract": "Let $d\\geq1$, $Λ>0$, and $ρ\\in\\mathbb{R}$, let $μ=\\mathcal{N}(0,Λ^{-1}\\mathrm{Id})$ on $\\mathbb{R}^{d}$, where $\\mathrm{Id}$ is the identity matrix, and let $ν$ be a probability measure whose support is a convex body $K$ of diameter $D=\\operatorname{diam}(K)$ and whose density is proportional to $e^{-W}$, where $\\mathrm{D}^{2}W\\succeq-ρ\\mathrm{Id}$. We prove that the Brenier map $T$ from $μ$ to $ν$ has Lipschitz constant $\\operatorname{Lip}(T)\\leq\\sqrt{Λ/(2π)}\\int_{-D/2}^{D/2}e^{ρt^{2}/2}\\,\\mathrm{d}t$, which is sharp in every fixed dimension. For log-concave targets, the optimal coefficient of $\\sqrtΛ\\,D$ is $1/\\sqrt{2π}$. For sources with finite second moment and density proportional to $e^{-V}$ for which $x\\mapsto\\frac12\\langle Qx,x\\rangle-V(x)$ is convex, where $Q$ is symmetric positive definite, and log-concave targets with compact support $K$, the Brenier potential $Φ$ satisfies $\\partial_{vv}Φ\\leq C_{1}\\sqrt{\\langle Qv,v\\rangle}\\,w_{K}(v)$ in the sense of distributions, where $w_{K}(v):=\\sup_{y\\in K}\\langle y,v\\rangle-\\inf_{y\\in K}\\langle y,v\\rangle$ is the width of $K$ along $v\\in\\mathbb{R}^{d}$ and $C_{1}<0.463$. This removes Kolesnikov's $\\sqrt d$ loss. For log-concave sources with $\\mathrm{D}^{2}V\\preceqΛ\\mathrm{Id}$ and the target $ν$, the Brenier map $T$ satisfies $\\operatorname{Lip}(T)^{2}\\leqΛ\\int_{0}^{D^{2}/4}e^{ρt}\\,\\mathrm{d}t$, so $\\operatorname{Lip}(T)$ has the optimal exponential rate $1/8$ in $ρD^{2}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.15906",
  "author_total": 1,
  "authors": [
    {
      "name": "Maja Gwozdz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3sty8309fcxda.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.15906",
  "fulltext_url": null,
  "id": "3sty8309fcxda",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.15906",
  "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.AP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7203686241",
  "pdf_url": "https://arxiv.org/pdf/2608.15906",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3sty8309fcxda.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:13Z",
      "native_id": "2608.15906",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.15906"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Dimension-Free Lipschitz Bounds for Brenier Maps to Compact Log-Concave Targets",
  "title_text": "Dimension-Free Lipschitz Bounds for Brenier Maps to Compact Log-Concave Targets",
  "type": "preprint",
  "updated_at": "2026-11-03T12:21:13Z",
  "url": "https://research.draconitic.com/paper/3sty8309fcxda",
  "venue": "arXiv",
  "versions": []
}