{
  "abstract": "Let $K$ be a convex body (a non-empty compact convex set) in $n$-dimensional Euclidean space. A set $B$ is called a barrier (or an `opaque set') for $K$ if every line that intersects $K$, also intersects $B$. Although this concept was introduced more than a century ago, the barrier with minimal surface area for a given set $K$ is still unknown, even in the two-dimensional case. A classical lower bound by Jones states that the surface area $S(B)$ of a sufficiently regular barrier $B$ is at least $S(\\partial K)/2$, half the surface area of the boundary of $K$. We will extend a known stability version for $n=2$ to arbitrary dimensions: if $S(B)-S(\\partial K)/2$ is small, then the orientation measure of $B$ is close to the surface area measure of a symmetrization of $K$. For instance, if $K$ is the unit cube in 3D, most of the points of a barrier with surface area close to $3$ must have almost axis parallel normals. One of the main contributions of the paper is the new concept of weak barriers, which only encodes orientation information of a barrier, disregarding the relative positions of its parts. We characterize weak barriers geometrically in terms of the convexification of $B$. Convex geometric tools then allow one to quantify the above mentioned stability for weak barriers in all dimensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.13449",
  "author_total": 1,
  "authors": [
    {
      "name": "Markus Kiderlen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4xkt50cwp0j2e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.13449",
  "fulltext_url": null,
  "id": "4xkt50cwp0j2e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.13449",
  "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",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7161164305",
  "pdf_url": "https://arxiv.org/pdf/2605.13449",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4xkt50cwp0j2e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2605.13449",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.13449"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Stability for barriers of n-dimensional convex bodies with surface area close to Jones' bound",
  "title_text": "Stability for barriers of n-dimensional convex bodies with surface area close to Jones' bound",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/4xkt50cwp0j2e",
  "venue": "arXiv",
  "versions": []
}