{"paper":{"title":"Around the A.D. Alexandrov's theorem on a characterization of a sphere","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Victor Alexandrov","submitted_at":"2012-12-20T14:20:38Z","abstract_excerpt":"This is a survey paper on various results relates to the following theorem first proved by A.D. Alexandrov: \\textit{Let $S$ be an analytic convex sphere-homeomorphic surface in $\\mathbb R^3$ and let $k_1(\\boldsymbol{x})\\leqslant k_2(\\boldsymbol{x})$ be its principal curvatures at the point $\\boldsymbol{x}$. If the inequalities $k_1(\\boldsymbol{x})\\leqslant k\\leqslant k_2(\\boldsymbol{x})$ hold true with some constant $k$ for all $\\boldsymbol{x}\\in S$ then $S$ is a sphere.} The imphases is on a result of Y. Martinez-Maure who first proved that the above statement is not valid for convex $C^2$-su"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.5047","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"}