{"paper":{"title":"One-dimensional projective structures, convex curves and the ovals of Benguria & Loss","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.SP"],"primary_cat":"math.DG","authors_text":"Jacob Bernstein, Thomas Mettler","submitted_at":"2014-03-31T14:00:35Z","abstract_excerpt":"Benguria and Loss have conjectured that, amongst all smooth closed curves of length $2\\pi$ in the plane, the lowest possible eigenvalue of the operator $L=-\\Delta+\\kappa^2$ was one. They observed that this value was achieved on a two-parameter family, $\\mathcal{O}$, of geometrically distinct ovals containing the round circle and collapsing to a multiplicity-two line segment. We characterize the curves in $\\mathcal{O}$ as absolute minima of two related geometric functionals. We also discuss a connection with projective differential geometry and use it to explain the natural symmetries of all th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.8000","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"}