Mean-field bounds for Poisson-Boolean percolation

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