Nonoptimality of constant radii in high dimensional continuum percolation

Consider a Boolean model . in Rd. The centers are given by a homogeneous Poisson point process with intensity . and the radii of distinct balls are i.i.d. with common distribution .. The critical covered volume is the proportion of space covered by . when the intensity . is critical for percolation. Previous numerical simulations and heuristic arguments suggest that the critical covered volume may be minimal when . is a Dirac measure. In this paper, we prove that it is not the case in sufficiently high dimension.