  Lower bounds for variances are often needed to derive central limit theorems.
In this paper, we establish a lower bound for the variance of Poisson
functionals that uses the difference operator of Malliavin calculus. Poisson
functionals, i.e. random variables that depend on a Poisson process, are
frequently studied in stochastic geometry. We apply our lower variance bound to
statistics of spatial random graphs, the $L^p$ surface area of random polytopes
and the volume of excursion sets of Poisson shot noise processes. Thereby we do
not only bound variances from below but also show positive definiteness of
asymptotic covariance matrices and provide associated results on the
multivariate normal approximation.
