{"paper":{"title":"Some approximation problems in semi-algebraic geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.OC"],"primary_cat":"math.AG","authors_text":"Malgorzata Stawiska, Shmuel Friedland","submitted_at":"2014-12-10T02:06:45Z","abstract_excerpt":"In this paper we deal with a best approximation of a vector with respect to a closed semi-algebraic set $C$ in the space $\\mathbb{R}^n$ endowed with a semi-algebraic norm $\\nu$. Under additional assumptions on $\\nu$ we prove semi-algebraicity of the set of points of unique approximation and other sets associated with the distance to $C$. For $C$ irreducible algebraic we study the critical point correspondence and introduce the $\\nu$- distance degree, generalizing the notion appearing in \\cite{DHOST} for the Euclidean norm. We discuss separately the case of the $\\ell^p$ norm ($p>1$)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.3178","kind":"arxiv","version":3},"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"}