{"paper":{"title":"Tight Bounds for Approximate Carath\\'eodory and Beyond","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","math.OC"],"primary_cat":"cs.DS","authors_text":"Adrian Vladu, Renato Paes Leme, Sam Chiu-wai Wong, Vahab Mirrokni","submitted_at":"2015-12-29T05:06:23Z","abstract_excerpt":"We give a deterministic nearly-linear time algorithm for approximating any point inside a convex polytope with a sparse convex combination of the polytope's vertices. Our result provides a constructive proof for the Approximate Carath\\'{e}odory Problem, which states that any point inside a polytope contained in the $\\ell_p$ ball of radius $D$ can be approximated to within $\\epsilon$ in $\\ell_p$ norm by a convex combination of only $O\\left(D^2 p/\\epsilon^2\\right)$ vertices of the polytope for $p \\geq 2$. We also show that this bound is tight, using an argument based on anti-concentration for th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.08602","kind":"arxiv","version":1},"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"}