Local inversion of a class of piecewise regular maps

This paper provides sufficient conditions for any map $L$, that is strongly piecewise linear relatively to a decomposition of $R^k$ in admissible cones, to be invertible. Namely, via a degree theory argument, we show that when there are at most four convex pieces (or three pieces with at most a non convex one), the map is invertible. Examples show that the result cannot be plainly extended to a greater number of pieces. Our result is obtained by studying the structure of strongly piecewise linear maps. We then extend the results to the $PC^1$ case.