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Let Gr_{n,m} denote the Grassmann manifold of m-dimensional linear subspaces of R^n and consider the projection distance d_p(W_1,W_2) := ||Pi_{W_1} - Pi_{W_2}|| (spectral norm) between W_1 and W_2 in Gr_{n,m}, where Pi_{W_i} denotes the orthogonal projection onto W_i. We call C_G(W) := max {d_p(W,W')^{-1} | W' \\in Sigma_m} the Grassmann condition number of W in Gr_{n,m}, where the set of ill-posed instances Sigma_m su"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1105.4049","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2011-05-20T09:32:28Z","cross_cats_sorted":[],"title_canon_sha256":"f105407ba71cdf70ec2f223cf4b5dbd8731bf92e531238dc951178d028592fbb","abstract_canon_sha256":"fdf5ac9b19404dfb6c0f070371c370b9eca06c7427e6a4fe890a0283222327c7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:40.505052Z","signature_b64":"Rl0/gfAQk1+vQGVwDQtQd2J8qw4UROpSI87g19hz/OqTmWVXN0iXcj1pHa//jDZvnQGNa7nQ/5b4LfU0eMIjAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"508914e459e20ecb5590e187bc8f5160d2b04647a564bbd9590e152358e78710","last_reissued_at":"2026-05-18T03:41:40.504233Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:40.504233Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A coordinate-free condition number for convex programming","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Dennis Amelunxen, Peter B\\\"urgisser","submitted_at":"2011-05-20T09:32:28Z","abstract_excerpt":"We introduce and analyze a natural geometric version of Renegar's condition number R for the homogeneous convex feasibility problem associated with a regular cone C subseteq R^n. Let Gr_{n,m} denote the Grassmann manifold of m-dimensional linear subspaces of R^n and consider the projection distance d_p(W_1,W_2) := ||Pi_{W_1} - Pi_{W_2}|| (spectral norm) between W_1 and W_2 in Gr_{n,m}, where Pi_{W_i} denotes the orthogonal projection onto W_i. We call C_G(W) := max {d_p(W,W')^{-1} | W' \\in Sigma_m} the Grassmann condition number of W in Gr_{n,m}, where the set of ill-posed instances Sigma_m su"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.4049","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":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1105.4049","created_at":"2026-05-18T03:41:40.504366+00:00"},{"alias_kind":"arxiv_version","alias_value":"1105.4049v2","created_at":"2026-05-18T03:41:40.504366+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.4049","created_at":"2026-05-18T03:41:40.504366+00:00"},{"alias_kind":"pith_short_12","alias_value":"KCERJZCZ4IHM","created_at":"2026-05-18T12:26:32.869790+00:00"},{"alias_kind":"pith_short_16","alias_value":"KCERJZCZ4IHMWVMQ","created_at":"2026-05-18T12:26:32.869790+00:00"},{"alias_kind":"pith_short_8","alias_value":"KCERJZCZ","created_at":"2026-05-18T12:26:32.869790+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KCERJZCZ4IHMWVMQ4GD3ZD2RMD","json":"https://pith.science/pith/KCERJZCZ4IHMWVMQ4GD3ZD2RMD.json","graph_json":"https://pith.science/api/pith-number/KCERJZCZ4IHMWVMQ4GD3ZD2RMD/graph.json","events_json":"https://pith.science/api/pith-number/KCERJZCZ4IHMWVMQ4GD3ZD2RMD/events.json","paper":"https://pith.science/paper/KCERJZCZ"},"agent_actions":{"view_html":"https://pith.science/pith/KCERJZCZ4IHMWVMQ4GD3ZD2RMD","download_json":"https://pith.science/pith/KCERJZCZ4IHMWVMQ4GD3ZD2RMD.json","view_paper":"https://pith.science/paper/KCERJZCZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1105.4049&json=true","fetch_graph":"https://pith.science/api/pith-number/KCERJZCZ4IHMWVMQ4GD3ZD2RMD/graph.json","fetch_events":"https://pith.science/api/pith-number/KCERJZCZ4IHMWVMQ4GD3ZD2RMD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KCERJZCZ4IHMWVMQ4GD3ZD2RMD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KCERJZCZ4IHMWVMQ4GD3ZD2RMD/action/storage_attestation","attest_author":"https://pith.science/pith/KCERJZCZ4IHMWVMQ4GD3ZD2RMD/action/author_attestation","sign_citation":"https://pith.science/pith/KCERJZCZ4IHMWVMQ4GD3ZD2RMD/action/citation_signature","submit_replication":"https://pith.science/pith/KCERJZCZ4IHMWVMQ4GD3ZD2RMD/action/replication_record"}},"created_at":"2026-05-18T03:41:40.504366+00:00","updated_at":"2026-05-18T03:41:40.504366+00:00"}