Local inversion of planar maps with nice nondifferentiability structure

TL;DR

This paper establishes sufficient conditions for invertibility of piecewise linear maps in the plane with up to four slices, highlighting the importance of convexity and the limitations of extending these results to more slices.

Contribution

It provides a new invertibility criterion for planar piecewise linear maps based on slicing and convexity assumptions, combining linear algebra and topology.

Findings

Invertibility guaranteed for maps with up to four slices under certain conditions

Convexity of slices is essential for invertibility when four slices are involved

The results do not extend straightforwardly to more than four slices

Abstract

When the plane is pie-sliced in $n\leq 4$ parts (with nonempty interior and common vertex at the origin) our main result provides a sufficient condition for any map $L$, that is continuous and piecewise linear relatively to this slicing, to be invertible. Some examples show that the assumptions of the theorem cannot be relaxed too much. In particular, convexity of the slices cannot be dropped altogether when $n=4$. This result cannot be plainly extended to a greater number of slices. Our result is proved by a combination of linear algebra and topological arguments.