{
  "title": "Equality in the Fradelizi–Paouris–Schütt second-moment inequality",
  "upload_type": "publication",
  "publication_type": "preprint",
  "description": "<p>Fradelizi, Paouris, and Schütt proved a sharp lower bound for the normalized second moment of a convex body whose extreme points have Euclidean norm at least r, and characterized equality for polytopes. This note proves their conjectured characterization for arbitrary convex bodies: equality in C₂(K) ≥ [r²+(n+1)‖g_K‖²]/(n+2) holds exactly for nondegenerate simplices with all vertices of norm r. The proof covers every dimension n≥1 and resolves the mathematical question catalogued as OWR-4136-011.</p><p>The four-page proof uses an almost-everywhere extreme-point simplex partition and an exact nonnegative deficit identity. A retained-pair estimate supplies a second finite-approximation route. The deposit includes LaTeX source, an exact standard-library Python verifier, independent AI proof audits, a source comparison, and a bounded priority search. All 298 finite checks across 18 cases pass; they do not replace the analytic proof. No earlier resolution was found in the searched literature, without certifying global priority. This is an unrefereed, AI-assisted preprint. Text is CC BY 4.0; code is MIT.</p>",
  "creators": [
    {
      "name": "Kriebel, Alec",
      "orcid": "0009-0001-9320-500X",
      "affiliation": "Independent researcher"
    }
  ],
  "publication_date": "2026-09-23",
  "version": "1.0.0",
  "access_right": "open",
  "license": "cc-by-4.0",
  "language": "eng",
  "keywords": [
    "convex geometry",
    "second moment",
    "equality cases",
    "extreme points",
    "simplex",
    "Fradelizi–Paouris–Schütt inequality",
    "OWR-4136-011"
  ],
  "related_identifiers": [
    {
      "identifier": "10.4153/CMB-2011-142-1",
      "relation": "references",
      "scheme": "doi"
    },
    {
      "identifier": "10.4171/OWR/2009/53",
      "relation": "references",
      "scheme": "doi"
    },
    {
      "identifier": "https://github.com/AlecKriebel/Math/tree/main/owr_4136_011_equality",
      "relation": "isSupplementedBy",
      "scheme": "url"
    },
    {
      "identifier": "https://aleckriebel.github.io/Math/papers/fps-equality/",
      "relation": "isDocumentedBy",
      "scheme": "url"
    }
  ],
  "notes": "Prepared with OpenAI Codex from a candidate supplied by the author from prior AI-assisted work, with separate AI verification agents. The inequality, simplex moment identity, and polytope equality case are due to Fradelizi, Paouris, and Schütt; the contribution is the general-convex-body equality classification. The audits are not external human peer review."
}