{"paper":{"title":"Regularity of optimal maps on the sphere: the quadratic cost and the reflector antenna","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gregoire Loeper","submitted_at":"2013-01-26T10:15:58Z","abstract_excerpt":"Building on the results of Ma, Trudinger and Wang \\cite{MTW}, and of the author \\cite{L5}, we study two problems of optimal transportation on the sphere: the first corresponds to the cost function $d^2(x,y)$, where $d(\\cdot,\\cdot)$ is the Riemannian distance of the round sphere; the second corresponds to the cost function $-\\log|x-y|$, it is known as the reflector antenna problem. We show that in both cases, the {\\em cost-sectional curvature} is uniformly positive, and establish the geometrical properties so that the results of \\cite{L5} and \\cite{MTW} can apply: global smooth solutions exist "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.6229","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"}