{"paper":{"title":"Whitney Extension Theorems for convex functions of the classes $C^1$ and $C^{1,\\omega}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.CA","authors_text":"Carlos Mudarra, Daniel Azagra","submitted_at":"2015-07-14T17:09:13Z","abstract_excerpt":"Let $C$ be a subset of $\\mathbb{R}^n$ (not necessarily convex), $f:C\\to\\mathbb{R}$ be a function, and $G:C\\to\\mathbb{R}^n$ be a uniformly continuous function, with modulus of continuity $\\omega$. We provide a necessary and sufficient condition on $f$, $G$ for the existence of a convex function $F\\in C^{1, \\omega}(\\mathbb{R}^n)$ such that $F=f$ on $C$ and $\\nabla F=G$ on $C$, with a good control of the modulus of continuity of $\\nabla F$ in terms of that of $G$. On the other hand, assuming that $C$ is compact, we also solve a similar problem for the class of $C^1$ convex functions on $\\mathbb{R"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.03931","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"}