{"paper":{"title":"Anti-Vietoris--Rips metric thickenings and Borsuk graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.AT","authors_text":"Alex Elchesen, Henry Adams, Michael Moy, Sucharita Mallick","submitted_at":"2025-03-11T20:00:59Z","abstract_excerpt":"For $X$ a metric space and $r\\ge 0$, the anti-Vietoris-Rips metric thickening $\\mathrm{AVR^m}(X;r)$ is the space of all finitely supported probability measures on $X$ whose support has spread at least $r$, equipped with an optimal transport topology. We study the anti-Vietoris-Rips metric thickenings of spheres. We have a homeomorphism $\\mathrm{AVR^m}(S^n;r) \\cong S^n$ for $r > \\pi$, a homotopy equivalence $\\mathrm{AVR^m}(S^n;r) \\simeq \\mathbb{RP}^{n}$ for $\\frac{2\\pi}{3} < r \\le \\pi$, and contractibility $\\mathrm{AVR^m}(S^n;r) \\simeq *$ for $r=0$. For an $n$-dimensional compact Riemannian man"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.08862","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2503.08862/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"}