{"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"}