{"paper":{"title":"On the Homothety Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Deping Ye, Elisabeth M. Werner","submitted_at":"2009-11-03T17:38:58Z","abstract_excerpt":"Let $K$ be a convex body in $\\bbR^n$ and $\\d>0$. The homothety conjecture asks: Does $K_{\\d}=c K$ imply that $K$ is an ellipsoid? Here $K_{\\d}$ is the (convex) floating body and $c$ is a constant depending on $\\d$ only. In this paper we prove that the homothety conjecture holds true in the class of the convex bodies $B^n_p$, $1\\leq p\\leq \\infty$, the unit balls of $l_p^n$; namely, we show that $(B^n_p)_{\\d} = c B^n_p$ if and only if $p=2$. We also show that the homothety conjecture is true for a general convex body $K$ if $\\d$ is small enough. This improvs earlier results by Sch\\\"utt and Werne"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.0642","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":""},"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"}