{
  "abstract": "Let $Λ\\subseteq \\mathbb{R}^n$ be a full rank lattice with the property that $\\det Γ\\ge 1$ for all sublattices $Γ\\subseteq Λ^*$. We prove that there exists a symmetric random vector $X \\in \\mathbb{R}^n$ with a smooth, strictly positive density $f : \\mathbb{R}^n \\to (0, \\infty)$ satisfying that (1) $f$ is $α$-strongly log-concave, where $α= 2π/e$, and (2) $X + Λ$ is uniform on $\\mathbb{R}^n/Λ$. Moreover, the existence of such a density characterizes this property up to a universal dilation. There are a few consequences of the above. We list some. First, for such lattices we can bound the covering radius \\[ μ(Λ, K) = \\max_{x\\in\\mathbb{R}^n}\\min_{λ\\inΛ} \\left\\|x-λ\\right\\|_K \\le \\sqrt{\\frac{2e}π} \\cdot \\mathbb{E}\\,\\left\\|G\\right\\|_K \\] of a convex body $K \\subseteq \\mathbb{R}^n$ containing $0$ in its interior, where $G \\sim N(0,\\mathrm{Id})$. This implies an $O(n \\sqrt{\\log n})$ bound for the flatness constant and an $O(\\log^{3/2} n)$ bound for the subspace flatness constant. We also compare the entire ordered spectrum of the associated flat torus on $Λ^*$ with that of a Gaussian, yielding an alternative proof of the reverse Minkowski theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29238",
  "author_total": 3,
  "authors": [
    {
      "name": "Brayden Letwin"
    },
    {
      "name": "Victor Reis"
    },
    {
      "name": "Thomas Rothvoss"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/91jgwt90yzt7d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29238",
  "fulltext_url": null,
  "id": "91jgwt90yzt7d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29238",
  "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.MG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7223243979",
  "pdf_url": "https://arxiv.org/pdf/2610.29238",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/91jgwt90yzt7d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.29238",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29238"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Strongly log-concave measures and lattices",
  "title_text": "Strongly log-concave measures and lattices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/91jgwt90yzt7d",
  "venue": "arXiv",
  "versions": []
}