Poissonian tessellations of the Euclidean space. An extension of a result of R.E. Miles

In 1973, R.E. Miles obtained an explicit characterization of the typical cell of the planar Poissonian tessellation, by means of the distributions of the indisc and the triangle circumscribed to the cell. In this Note, we propose a different proof using the classical Slivnyak formula for Poisson point processes. Not only the method is simple and rigorous, but it also extends R.E. Miles' result to any dimension d greater than or equal to 2. We deduce from it some other properties of the geometrical characteristics of the typical cell. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.