A note on surjectivity of piecewise affine mappings

TL;DR

This paper proves that a piecewise affine function's surjectivity can be established from the orientation of its active selection functions on unbounded sets, simplifying existing conditions for surjectivity.

Contribution

It shows surjectivity follows from coherent orientation on unbounded sets, extending classical results and providing a shorter proof for injective functions.

Findings

Surjectivity follows from orientation on unbounded sets

Injective piecewise affine functions are coherently oriented

Simplified proof of classical surjectivity condition

Abstract

A standard theorem in nonsmooth analysis states that a piecewise affine function $F:\mathbb R^n\rightarrow\mathbb R^n$ 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.