{"paper":{"title":"Quantitative stability of the intersection body operator near the ball, and the dynamical origin of the two--dimensional degeneracy","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.MG","authors_text":"S. Spektor","submitted_at":"2026-07-10T13:44:32Z","abstract_excerpt":"Let $\\IB$ denote the intersection body operator on star bodies in $\\R^n$. A recent theorem of Milman, Shabelman and Yehudayoff establishes that for $n\\ge 3$ the equation $\\IB^2 K = cK$ holds if and only if $K$ is a centered ellipsoid, thereby resolving the fixed--point problem for $\\IB^2$ and, as a consequence, the long--standing conjecture $\\IB K = cK \\Leftrightarrow K$ is a ball. We complement this qualitative rigidity with a \\emph{quantitative} analysis in a neighbourhood of the ball. Linearizing the associated shape dynamics on $L^2(\\Sph)$, we compute the full spectrum of the operator $\\IB"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.09412","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/2607.09412/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"}