{"paper":{"title":"Area of convex disks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Christopher Croke, Gregory R. Chambers, Haomin Wen, Yevgeny Liokumovich","submitted_at":"2017-01-23T19:15:17Z","abstract_excerpt":"This paper considers metric balls $B(p,R)$ in two dimensional Riemannian manifolds when $R$ is less than half the convexity radius. We prove that $Area(B(p,R)) \\geq \\frac{8}{\\pi}R^2$. This inequality has long been conjectured for $R$ less than half the injectivity radius. This result also yields the upper bound $\\mu_2(B(p,R)) \\leq 2(\\frac{\\pi}{2 R})^2$ on the first nonzero Neumann eigenvalue $\\mu_2$ of the Laplacian in terms only of the radius. This has also been conjectured for $R$ up to half the injectivity radius."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.06594","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"}