{"paper":{"title":"An Extension Theorem for convex functions of class $C^{1,1}$ on Hilbert spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Carlos Mudarra, Daniel Azagra","submitted_at":"2016-03-01T11:47:37Z","abstract_excerpt":"Let $\\mathbb{H}$ be a Hilbert space, $E \\subset \\mathbb{H}$ be an arbitrary subset and $f: E \\rightarrow \\mathbb{R}, \\: G: E \\rightarrow \\mathbb{H}$ be two functions. We give a necessary and sufficient condition on the pair $(f,G)$ for the existence of a \\textit{convex} function $F\\in C^{1,1}(\\mathbb{H})$ such that $F=f$ and $\\nabla F =G$ on $E$. We also show that, if this condition is met, $F$ can be taken so that $\\textrm{Lip}(\\nabla F) = \\textrm{Lip}(G)$. We give a geometrical application of this result, concerning interpolation of sets by boundaries of $C^{1,1}$ convex bodies in $\\mathbb{H"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.00241","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"}