{
  "abstract": "We prove a sharp Santaló-type inequality for the product of $V(φ)$ and $V$ of the Legendre transform of $φ$, where $φ$ is an even geometric convex function on $\\mathbb{R}^n$ and $V(φ)=\\int(1+φ)^{-(n+1)}$. For this size functional, the Legendre transform and functional polarity give the same volume, so the same inequality holds for both dualities. Through a projective correspondence, the problem is equivalent to a new Santaló-type inequality with respect to standard volume and standard polarity, on a class of not necessarily centered convex bodies. This class consists of convex bodies $K\\subset\\mathbb R^n\\times[-1,1]$ containing $\\{0\\}\\times[-1,1]$, for which every horizontal section is centrally symmetric about the distinguished vertical axis. We determine the sharp upper bound for $\\operatorname{vol}_{n+1}(K)\\operatorname{vol}_{n+1}(K^\\circ)$ in this class and classify all equality cases. When $n=1$, the extremizers are the disk and its horizontal linear images. When $n\\ge 2$, the ball is no longer a maximizer, and the extremizer is unique up to horizontal linear transformations and the reflection $t\\mapsto -t$. The symmetric Santaló inequality applied to horizontal sections, together with monotone transport, reduces the problem to a maximization problem for increasing curves in the square $[-1,1]^2$. We solve this problem using a global potential when $n=1$ and two Hamilton-Jacobi branches when $n\\ge 2$. Finally, in dimension one we provide a sharp family of functional Santaló inequalities for the Legendre duality, interpolating between the volume $V$ and the exponential volume $\\int \\exp(-φ)$, with centered quadratics as the only equality cases.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38041",
  "author_total": 1,
  "authors": [
    {
      "name": "Shiri Artstein-Avidan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dvt8y138x5jcw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38041",
  "fulltext_url": null,
  "id": "dvt8y138x5jcw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38041",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7214889244",
  "pdf_url": "https://arxiv.org/pdf/2609.38041",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dvt8y138x5jcw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.38041",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38041"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A sharp Santaló inequality in a slab with centrally symmetric sections",
  "title_text": "A sharp Santaló inequality in a slab with centrally symmetric sections",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/dvt8y138x5jcw",
  "venue": "arXiv",
  "versions": []
}