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

On Some fundamental aspects of Polyominoes on Random Voronoi Tilings