Asymmetric tropical distances and power diagrams

We investigate Voronoi diagrams with respect to an asymmetric tropical distance function, in particular for infinite point sets. These Voronoi diagrams turn out to be much better behaved than those arising from the standard tropical distance, which is symmetric. In particular, we show that the asymmetric tropical Voronoi diagrams may be seen as tropicalizations of power diagrams over fields of real Puiseux series. Our results are then applied to rational lattices and Laurent monomial modules.