Indistinguishability of cells for the ideal Poisson Voronoi tessellation

In this note, we resolve a question of D’Achille, Curien, Enriquez, Lyons, and Ünel by showing that the cells of the ideal Poisson Voronoi tessellation are indistinguishable. This follows from an application of the Howe-Moore theorem and a theorem of Meyerovitch about the nonexistence of thinnings of the Poisson point process. We also give an alternative proof of Meyerovitch’s theorem.