{"paper":{"title":"A note on Hurwitz's inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Agust\\'i Revent\\'os, Eduardo Gallego, Juli\\`a Cuf\\'i","submitted_at":"2017-04-04T10:29:26Z","abstract_excerpt":"Given a simple closed plane curve $\\Gamma$ of length $L$ enclosing a compact convex set $K$ of area $F$, Hurwitz found an upper bound for the isoperimetric deficit, namely $L^2-4\\pi F\\leq \\pi |F_{e}|$, where $F_{e}$ is the algebraic area enclosed by the evolute of $\\Gamma$.\n  In this note we improve this inequality finding strictly positive lower bounds for the deficit $\\pi|F_{e}|-\\Delta$, where $\\Delta=L^{2}-4\\pi F$. These bounds involve wether the visual angle of $\\Gamma$ or the pedal curve associated to $K$ with respect to the Steiner point of $K$ or the $\\mathcal{L}^{2}$ distance between $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.00944","kind":"arxiv","version":2},"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"}