  We prove a new class of inequalities, yielding bounds for the normal
approximation in the Wasserstein and the Kolmogorov distance of functionals of
a general Poisson process (Poisson random measure). Our approach is based on an
iteration of the classical Poincar\'e inequality, as well as on the use of
Malliavin operators, of Stein's method, and of an (integrated) Mehler's
formula, providing a representation of the Ornstein-Uhlenbeck semigroup in
terms of thinned Poisson processes. Our estimates only involve first and second
order differential operators, and have consequently a clear geometric
interpretation. In particular we will show that our results are perfectly
tailored to deal with the normal approximation of geometric functionals
displaying a weak form of stabilization, and with non-linear functionals of
Poisson shot-noise processes. We discuss two examples of stabilizing
functionals in great detail: (i) the edge length of the $k$-nearest neighbour
graph, (ii) intrinsic volumes of $k$-faces of Voronoi tessellations. In all
these examples we obtain rates of convergence (in the Kolmogorov and the
Wasserstein distance) that one can reasonably conjecture to be optimal, thus
significantly improving previous findings in the literature. As a necessary
step in our analysis, we also derive new lower bounds for variances of Poisson
functionals.
