Grain rotations and distortions in the asymptotic variance of vacancy of the Boolean model

We consider the asymptotic variance of vacancy (AVV) in the high intensity small-grain Boolean model. Subjecting the grains to rotations or, more generally, linear distortions gives rise to a function which maps distortion distributions to the AVV of the corresponding Boolean model. We mainly study continuity properties of this function, where we use the L1 Wasserstein metric on distortion distributions. An important role in the formulation and derivation of our results is played by notions of symmetry commonly used in multivariate analysis and stochastic simulation, such as conjugation-invariance and group models.