  We establish a regularity result for optimal sets of the isoperimetric
problem with double density under mild ($\alpha$-)H\"older regularity
assumptions on the density functions. Our main Theorem improves some previous
results and allows to reach in any dimension the regularity class
$C^{1,\frac{\alpha}{2-\alpha}}$. This class is indeed the optimal one for local
minimizers of variational functionals with an integrand that depends
$\alpha$-H\"older continuous on the minimizer itself, and as such can (the
boundary of) the isoperimetric set be locally written (with additional
constraint).
