{"paper":{"title":"Borsuk-Ulam type theorem for Stiefel manifolds and orthogonal mass partitions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A Borsuk-Ulam theorem generalized to Stiefel manifolds yields bounds on d guaranteeing k mutually orthogonal hyperplanes whose any n subset equally partitions each of m given measures in R^d.","cross_cats":["math.CO","math.MG"],"primary_cat":"math.AT","authors_text":"Oleg R. Musin","submitted_at":"2026-03-19T07:07:43Z","abstract_excerpt":"A generalization of the Borsuk-Ulam theorem to Stiefel manifolds is considered. This theorem is applied to derive bounds on $d$ that guarantee-for a given set of $m$ measures in $\\mathbb{R}^d$-the existence of $k$ mutually orthogonal hyperplanes, any $n$ of which partition each of the measures into $2^n$ equal parts. If $n=k$, the result corresponds to the bound obtained in [14], but with the stronger conclusion that the hyperplanes are mutually orthogonal."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"A generalization of the Borsuk-Ulam theorem to Stiefel manifolds is considered. This theorem is applied to derive bounds on d that guarantee-for a given set of m measures in R^d-the existence of k mutually orthogonal hyperplanes, any n of which partition each of the measures into 2^n equal parts.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The measures are continuous (or absolutely continuous) probability measures on R^d so that the topological configuration space and test maps satisfy the conditions needed for the generalized Borsuk-Ulam theorem to apply.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A Borsuk-Ulam type theorem for Stiefel manifolds yields bounds on d guaranteeing k mutually orthogonal hyperplanes that partition m measures into equal parts.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A Borsuk-Ulam theorem generalized to Stiefel manifolds yields bounds on d guaranteeing k mutually orthogonal hyperplanes whose any n subset equally partitions each of m given measures in R^d.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c1d30b78f72fed5d41cf37e4fde3f041741984688881910175c7fd58121a6091"},"source":{"id":"2603.18550","kind":"arxiv","version":3},"verdict":{"id":"f9539149-77e6-4f44-bbfe-c34f49d87f34","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T08:58:05.969182Z","strongest_claim":"A generalization of the Borsuk-Ulam theorem to Stiefel manifolds is considered. This theorem is applied to derive bounds on d that guarantee-for a given set of m measures in R^d-the existence of k mutually orthogonal hyperplanes, any n of which partition each of the measures into 2^n equal parts.","one_line_summary":"A Borsuk-Ulam type theorem for Stiefel manifolds yields bounds on d guaranteeing k mutually orthogonal hyperplanes that partition m measures into equal parts.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The measures are continuous (or absolutely continuous) probability measures on R^d so that the topological configuration space and test maps satisfy the conditions needed for the generalized Borsuk-Ulam theorem to apply.","pith_extraction_headline":"A Borsuk-Ulam theorem generalized to Stiefel manifolds yields bounds on d guaranteeing k mutually orthogonal hyperplanes whose any n subset equally partitions each of m given measures in R^d."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.18550/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"}