{"paper":{"title":"Inscribed Polygons that Characterize Inner Product Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Carlos Ben\\'itez, Diego Y\\'a\\~nez, Pedro Mart\\'in","submitted_at":"2017-07-28T09:59:06Z","abstract_excerpt":"Let $X$ be a real normed space with unit sphere S. We prove that $X$ is an inner product space if and only if there exists a real number $\\rho=\\sqrt{(1+\\cos\\frac{2k\\pi}{2m+1})/2}$, $(k=1,2,\\ldots , m ;\\:m=1,2,\\ldots)$, such that every chord of $S$ that supports $\\rho S$ touches $\\rho S$ at its middle point. If this condition holds, then every point $u\\in S$ is a vertex of a regular polygon that is inscribed in $S$ and circumscribed about $\\rho S$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.09171","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"}