2016/12/14 by Jacek Gałęski, Gałęski, Jacek
Mathematics · #28A75 #57N20 #Advanced Differential Equations and Dynamical Systems #Advanced Optimization Algorithms Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:28A75 #msc:57N20
paper · pdf · doi:10.48550/arxiv.1612.04578
arxiv created 2016/12/14 · openalex publication_date 2016/12/14 · arxiv updated 2016/12/15 · openalex created_date 2017/05/26 · openalex updated_date 2026/07/28
The purpose of this article is to prove a generalisation of the Besicovitch-Federer projection theorem about a characterisation of rectifiable and unrectifiable sets in terms of their projections. For an m-unrectifiable set Σ⊂ℝn having finite Hausdorff measure and ε>0, we prove that for a mapping f\inC1(U,ℝn) having constant, equal to m, rank of the Jacobian matrix there exists a mapping fε whose rank of the Jacobian matrix is also constant, equal to m, such that ‖fε-f‖C1<ε and Hm(fε(Σ))=0. We derive it as a consequence of the Besicovitch-Federer theorem stating that the Hm measure of a generic projection of an m-unrectifiable set Σ onto an m-dimensional plane is equal to zero.