{"paper":{"title":"Smooth convex extensions of convex functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.DG"],"primary_cat":"math.CA","authors_text":"Carlos Mudarra, Daniel Azagra","submitted_at":"2015-01-21T16:48:33Z","abstract_excerpt":"Let $C$ be a compact convex subset of $\\mathbb{R}^n$, $f:C\\to\\mathbb{R}$ be a convex function, and $m\\in\\{1, 2, ..., \\infty\\}$. Assume that, along with $f$, we are given a family of polynomials satisfying Whitney's extension condition for $C^m$, and thus that there exists $F\\in C^{m}(\\mathbb{R}^n)$ such that $F=f$ on $C$. It is natural to ask for further (necessary and sufficient) conditions on this family of polynomials which ensure that $F$ can be taken to be convex as well. We give a satisfactory solution to this problem in the case $m=\\infty$, and also less satisfactory solutions in the ca"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.05226","kind":"arxiv","version":8},"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"}