2017/07/27 by Manuel Radons, Radons, Manuel
Mathematics · #15A39 #49J52 #57Q99 #FOS: Mathematics #Optimization and Control (math.OC) #math.OC #msc:15A39 #msc:49J52 #msc:57Q99
paper · pdf · doi:10.48550/arxiv.1707.08786
4 Pages, 1 Figure
arxiv created 2018/01/22 · arxiv updated 2018/01/23
A standard theorem in nonsmooth analysis states that a piecewise affine function F:\mathbb Rn→\mathbb Rn is surjective if it is coherently oriented in that the linear parts of its selection functions all have the same nonzero determinant sign. In this note we prove that surjectivity already follows from coherent orientation of the selection functions which are active on the unbounded sets of a polyhedral subdivision of the domain corresponding to F. A side bonus of the argumentation is a short proof of the classical statement that an injective piecewise affine function is coherently oriented.