{
  "abstract": "We study $\\ell_0$ isoperimetry for a convex body $K\\subset \\mathbb{R}^n$, $n\\ge2$. For a Borel set $S\\subset K$, let $\\partial_0^K S$ be the set of points in $K \\setminus S$ that can be reached from $S$ by changing at most one coordinate (i.e. the $\\ell_0$ boundary of $S$). Suppose that, for some unconditional convex body $Q \\subset \\mathbb{R}^n$, numbers $r,R>0$, and possibly different centers $x_0,y_0$, \\[ x_0+rQ \\subset K\\subset y_0+RQ. \\] Writing $s=\\text{vol}(S)/\\text{vol}(K)$, we prove that whenever $0<s \\le 1/2$, \\[ \\frac{\\text{vol}(\\partial_0^K S)}{\\text{vol}(S)} \\ge \\frac{cr}{nR} \\min\\left\\{1,\\frac{\\log(e/s)}{n}\\right\\}, \\] where $c > 0$ is an absolute constant. Consequently, the associated $\\ell_0$-isoperimetric coefficient is at least $cr/(n^2R)$. Previous direct lower bounds were only known for $\\ell_2$ and $\\ell_\\infty$ regularity whereas our lower bound holds directly for any $Q$-regularity, where $Q$ is an unconditional convex body. Compared to $\\ell_2$ and $\\ell_\\infty$ regularity, our lower bound result improves upon the previously best known lower bounds, for any $s$, by a factor of $n$. As an application of our result, we give improved mixing time bounds for the Coordinate Hit and Run walk (CHAR). Our proof of the lower bound is based on a modification of the method of canonical paths applied to a continuous Hamming graph over our convex body. Our construction of canonical paths can be viewed as a suitable coordinate discretization of certain mass transport maps from $S$ to $S^c$. We also give complementary upper-bounds for any $Q$-regularity, with an overall factor of $n$ gap between the two.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27854",
  "author_total": 1,
  "authors": [
    {
      "name": "Manuel Fernandez V"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0n54t4nr062s3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27854",
  "fulltext_url": null,
  "id": "0n54t4nr062s3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27854",
  "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",
      "cs.CG"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "arxiv",
      "math.ST"
    ],
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      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
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  "openalex": "W7204776702",
  "pdf_url": "https://arxiv.org/pdf/2608.27854",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0n54t4nr062s3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.27854",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27854"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Improved $\\ell_0$-Isoperimetry for Convex Bodies via Mass Transport",
  "title_text": "Improved ℓ_0-Isoperimetry for Convex Bodies via Mass Transport",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:34Z",
  "url": "https://research.draconitic.com/paper/0n54t4nr062s3",
  "venue": "arXiv",
  "versions": []
}