Mean-field bounds for Poisson-Boolean percolation

We establish the mean-field bounds γ≥1, δ≥2 and △≥2 on critical exponents of Poisson-Boolean continuum percolation model under a moment condition radii; these were previously known only in special case fixed radii (in γ), or not at all δ △). deduce as consequences bound β≤1, recently established same [8], using relative entropy method introduced by authors previous work [7].