{
  "abstract": "We characterize the fixed points of the operator $Γ_pΠ_p^\\ast$ for $n\\geq 3$ and $1<p<n$. More precisely, we prove that a convex body $K\\in\\mathcal K_o^n$ satisfies \\[Γ_pΠ_p^\\ast K=cK\\] for some $c>0$ if and only if $K$ is an origin-centered ellipsoid, thereby settling the Lutwak--Yang--Zhang fixed-point conjecture in this range. Our proof is based on a variational analysis along linear reflection shadow systems. To address the nonlinear structure of the $L_p$ setting, we introduce the $L_p$-Projection Rolodex, which provides a dimensional reduction of the volume of the polar $L_p$-projection body to weighted lower-dimensional sectional functionals. A suitable change of variables, together with Ball's harmonic Prékopa--Leindler inequality, yields the convexity needed along the shadow system. Under the fixed-point condition, a first-variation identity then forces $\\operatorname{vol}_n(Π_p^\\ast K_t)$ to remain constant throughout the deformation. The rigidity statement follows from the equality characterization under Steiner symmetrization.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.25666",
  "author_total": 2,
  "authors": [
    {
      "name": "Youjiang Lin"
    },
    {
      "name": "Sudan Xing"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ebeb48zvsmq2p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.25666",
  "fulltext_url": null,
  "id": "ebeb48zvsmq2p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.25666",
  "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.FA"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7162408933",
  "pdf_url": "https://arxiv.org/pdf/2605.25666",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ebeb48zvsmq2p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2605.25666",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.25666"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Fixed Point Rigidity of the Operator $Γ_pΠ_p^\\ast$ and the LYZ Conjecture",
  "title_text": "Fixed Point Rigidity of the Operator Γ_pΠ_p^∗ and the LYZ Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/ebeb48zvsmq2p",
  "venue": "arXiv",
  "versions": []
}