{"paper":{"title":"Distortion from spheres into Euclidean spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Any map from the round n-sphere of radius r into Euclidean n-space must additively distort distances by at least πr divided by 1 plus the square root of 1 minus a term that depends on the parity of n.","cross_cats":["math.GN"],"primary_cat":"math.MG","authors_text":"James Dibble","submitted_at":"2025-04-03T04:50:47Z","abstract_excerpt":"Any function from a round $n$-dimensional sphere of radius $r$ into $n$-dimensional Euclidean space must distort the metric additively by at least $\\displaystyle \\frac{\\pi r}{1 + \\sqrt{1 - \\frac{2}{n+2}}}$ if $n$ is even and $\\displaystyle \\frac{\\pi r}{1 + \\sqrt{1 - \\frac{2(n+2)}{(n+1)(n+3)}}}$ if $n$ is odd. This is proved using a fixed-point theorem of Granas that generalizes the classical theorem of Borsuk-Ulam to set-valued functions."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Any function from a round n-dimensional sphere of radius r into n-dimensional Euclidean space must distort the metric additively by at least πr / (1 + sqrt(1 - 2/(n+2))) if n even and πr / (1 + sqrt(1 - 2(n+2)/((n+1)(n+3)))) if n odd.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The set-valued map constructed from the sphere and the candidate distortion function satisfies the hypotheses (upper semicontinuity, convex values, etc.) of Granas' fixed-point theorem, as invoked in the proof.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Any map from the round n-sphere of radius r to R^n must distort distances additively by at least a positive constant depending on n and r, proved via Granas' set-valued fixed-point theorem.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Any map from the round n-sphere of radius r into Euclidean n-space must additively distort distances by at least πr divided by 1 plus the square root of 1 minus a term that depends on the parity of n.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"6228eaf61a6e70fbc09e5bab16009cdcf60e01e4e85ce4a057c4c13b72ef810d"},"source":{"id":"2504.02276","kind":"arxiv","version":4},"verdict":{"id":"a27a8395-cd15-4a8e-9df6-b76761b8c0a8","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-22T21:54:27.067113Z","strongest_claim":"Any function from a round n-dimensional sphere of radius r into n-dimensional Euclidean space must distort the metric additively by at least πr / (1 + sqrt(1 - 2/(n+2))) if n even and πr / (1 + sqrt(1 - 2(n+2)/((n+1)(n+3)))) if n odd.","one_line_summary":"Any map from the round n-sphere of radius r to R^n must distort distances additively by at least a positive constant depending on n and r, proved via Granas' set-valued fixed-point theorem.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The set-valued map constructed from the sphere and the candidate distortion function satisfies the hypotheses (upper semicontinuity, convex values, etc.) of Granas' fixed-point theorem, as invoked in the proof.","pith_extraction_headline":"Any map from the round n-sphere of radius r into Euclidean n-space must additively distort distances by at least πr divided by 1 plus the square root of 1 minus a term that depends on the parity of n."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2504.02276/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"}