  Consider a Voronoi tiling of the Euclidean space based on a realization of a
inhomogeneous Poisson random set. A Voronoi polyomino is a finite and connected
union of Voronoi tiles. In this paper we provide tail bounds for the number of
boxes that are intersected by a Voronoi polyomino, and vice-versa. These
results will be crucial to analyze self-avoiding paths, greedy polyominoes and
first-passage percolation models on Voronoi tilings and on the dual graph,
named the Delaunay triangulation.
