On the spectral function of the Poisson-Voronoi cells

Denote by phi (t) = Sigma (n greater than or equal to1) e(-lambda nt), t > 0, the spectral function related to the Dirichlet Laplacian for the typical cell e of a standard Poisson-Voronoi tessellation in R-d, d greater than o r equal to 2. We show that the expectation E phi (t), t > 0, is a functiona l of the convex hull of a standard d-dimensional Brownian bridge. This enab les us to study the asymptotic behaviour of E phi (t), when t --> 0(+), +in finity. In particular; we prove that in the two-dimensional case (d = 2) th e law of the first eigenvalue lambda (1) of C satisfies the asymptotic rela tion ln Ee(-t lambda1) similar to -t(1/2)4 root pi j(0), when t --> +infini ty, where j(0) is the first zero of the Bessel function J(0). (C) 2001 Acad emie des sciences/Editions scientifiques et medicales Elsevier SAS.