{"paper":{"title":"Global and fine approximation of convex functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.FA"],"primary_cat":"math.DG","authors_text":"Daniel Azagra","submitted_at":"2012-01-23T16:40:56Z","abstract_excerpt":"Let $U\\subseteq\\mathbb{R}^d$ be open and convex. We prove that every (not necessarily Lipschitz or strongly) convex function $f:U\\to\\mathbb{R}$ can be approximated by real analytic convex functions, uniformly on all of $U$. We also show that $C^0$-fine approximation of convex functions by smooth (or real analytic) convex functions on $\\mathbb{R}^d$ is possible in general if and only if $d=1$. Nevertheless, for $d\\geq 2$ we give a characterization of the class of convex functions on $\\mathbb{R}^d$ which can be approximated by real analytic (or just smoother) convex functions in the $C^0$-fine t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.4760","kind":"arxiv","version":6},"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"}