Euclidean models of first-passage percolation

We introduce a new family of first-passage percolation (FPP) models in the context of Poisson-Voronoi tesselations of ℝd. Compared to standard FPP on ℤd, these models have some technical complications but also have the advantage of statistical isotropy. We prove two almost sure results: a shape theorem (where isotropy implies an exact Euclidean ball for the asymptotic shape) and nonexistence of certain doubly infinite geodesics (where isotropy yields a stronger result than in standard FPP).