{"paper":{"title":"Total diameter and area of closed submanifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT","math.MG"],"primary_cat":"math.DG","authors_text":"Mohammad Ghomi, Ralph Howard","submitted_at":"2013-12-04T23:37:13Z","abstract_excerpt":"The total diameter of a closed planar curve $C\\subset R^2$ is the integral of its antipodal chord lengths. We show that this quantity is bounded below by twice the area of $C$. Furthermore, when $C$ is convex or centrally symmetric, the lower bound is twice as large. Both inequalities are sharp and the equality holds in the convex case only when $C$ is a circle. We also generalize these results to $m$ dimensional submanifolds of $R^n$, where the \"area\" will be defined in terms of the mod $2$ winding numbers of the submanifold about the $n-m-1$ dimensional affine subspaces of $R^n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.1384","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"}