{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-space-of-spaces-curvature-bounds-and","title":"The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces","arxiv_id":"1208.0434","date":"2012-08-02","proceeding":null,"authors":["Karl-Theodor Sturm"],"abstract":"Equipped with the L^2-distortion distance, the space \"X\" of all metric measure spaces (X,d,m) is proven to have nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are characterized in detail. Moreover, classes of semiconvex functionals and their gradient flows on \"X\" are presented.","url_abs":"https://arxiv.org/abs/1208.0434v1","url_pdf":"https://arxiv.org/pdf/1208.0434v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-space-of-spaces-curvature-bounds-and","repo_url":"https://github.com/trneedham/gromov-wasserstein-statistics","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1208.0434","atlas_url":"https://app.syntology.ai/?focus=1208.0434","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}