{"paper":{"title":"Pyramids and Extended Metric Measure Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.MG","authors_text":"Kohei Suzuki, Nicola Gigli, Simone Vincini","submitted_at":"2026-07-29T08:53:50Z","abstract_excerpt":"A pyramid is a generalisation of metric measure spaces (mm-spaces) introduced by M.~Gromov (Birkh\\\"auser 1999) to establish a geometric framework for measure-concentration problems. An extended metric measure space (emm-space) is another generalisation of mm-spaces introduced by Ambrosio--Gigli--Savar\\'e (Invent.~Math.~2014) to extend Sobolev calculus and optimal transport theory to a broader extent.\n  We prove that every pyramid has a representation by an emm-space through $1$-Lipschitz order. Furthermore, if pyramids are concentrated, this correspondence is unique up to isomorphism. This sho"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.26626","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.26626/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}