For a homogeneous Poisson point process in R^d we add a deterministic point to it. The Voronoi cell of this point is said to be the typical cell and its properties capture the properties of the "average" cell.  We derive sharp asymptotics  for geometric characteristics such as inradius, outradius, diameter, mean width, and the geometry of faces of the typical cell as d goes to infinity. 

We place n i.i.d. points uniformly on the unit sphere in d dimensions. Two points are connected iff they are at a distance of more than alpha(n) from another. As n goes to infinity we show how the chromatic number of this graph behaves in probability with respect to alpha(n).

For the Poisson-Voronoi tessellation in hyperbolic space with dimension at least three we color each cell black with probability p, independently from another. We show that the threshold for p  for the existence of an unique infinite component stays bounded away from 0 as the intensity of the Poisson point process converges to 0.

For the Poisson-Voronoi tessellation in Euclidean space we color each cell black with probability p, independently from another. The critical probability is the value for which above there can exist infinite connected black components. We give the asymptotic behavior of this critical probability as d goes to infinity. We also obtain the corresponding result for site percolation on the Poisson-Gabriel graph and show how the number of adjacent cells of the typical cell behaves as the dimension grows to infinity.