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Lower bounds for variances are often needed to derive central limit theorems.In this paper, we establish a lower bound for the variance of Poissonfunctionals that uses the difference operator of Malliavin calculus. Poissonfunctionals, i.e. random variables that depend on a Poisson process, arefrequently studied in stochastic geometry. We apply our lower variance bound tostatistics of spatial random graphs, the $L^p$ surface area of random polytopesand the volume of excursion sets of Poisson shot noise processes. Thereby we donot only bound variances from below but also show positive definiteness ofasymptotic covariance matrices and provide associated results on themultivariate normal approximation.

Lower bounds for variances of Poisson functionals