{"paper":{"title":"Fr\\'echet Barycenters and a Law of Large Numbers for Measures on the Real Line","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexey Kroshnin, Andrei Sobolevski","submitted_at":"2015-12-28T14:16:23Z","abstract_excerpt":"Endow the space $\\mathcal{P}(\\mathbb{R})$ of probability measures on $\\mathbb{R}$ with a transportation cost $J(\\mu, \\nu)$ generated by a translation-invariant convex cost function.\n  For a probability distribution on $\\mathcal{P}(\\mathbb{R})$ we formulate a notion of average with respect to this transportation cost, called here the Fr\\'echet barycenter, prove a version of the law of large numbers for Fr\\'echet barycenters, and briefly discuss the structure of $\\mathcal{P}(\\mathbb{R})$ related to the transportation cost $J$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.08421","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"}