{"paper":{"title":"Fixed Point Rigidity of the Operator $\\Gamma_p\\Pi_p^\\ast$ and the LYZ Conjecture","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Sudan Xing, Youjiang Lin","submitted_at":"2026-05-25T10:15:34Z","abstract_excerpt":"Motivated by the recent approach of Milman, Shabelman, and Yehudayoff \\cite{MilmanShabelmanYehudayoff2025}, we establish, for $p\\geq 1$, a complete characterization of the fixed points of the composition of the $L_p$-centroid operator and the polar $L_p$-projection operator. More precisely, we prove that if a convex body $K \\in \\mathcal{K}_o^n$ satisfies \\[\\Gamma_p \\Pi_p^* K = cK\\] for some constant $c>0$, then $K$ must be an ellipsoid. Conversely, ellipsoids are the only such fixed points of convex bodies up to dilation. This confirms a conjecture of Lutwak, Yang, and Zhang \\cite{LutwakYangZh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.25666","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.25666/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}