{"paper":{"title":"Piecewise linear approximation of smooth functions of two variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Joseph H.G. Fu, Ryan C. Scott","submitted_at":"2013-05-09T21:35:27Z","abstract_excerpt":"Given a piecewise linear (PL) function $p$ defined on an open subset of $\\R^n$, one may construct by elementary means a unique polyhedron with multiplicities $\\D(p)$ in the cotangent bundle $\\R^n\\times \\R^{n*}$ representing the graph of the differential of $p$. Restricting to dimension 2, we show that any smooth function $f(x,y)$ may be approximated by a sequence $p_1,p_2,\\dots$ of PL functions such that the areas of the $\\D(p_i)$ are locally dominated by the area of the graph of $df$ times a universal constant."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.2220","kind":"arxiv","version":2},"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"}