Although the main theme of this book is random geometric structures with invariance properties, such as stationarity or isotropy, we conclude with an outlook to some of the extensions that are possible without such assumptions. The invariance properties in previous chapters allowed us to employ integral geometric formulas for obtaining results on geometric mean values. Our set-up followed also the historical development of the field, where from the beginning stationarity and isotropy seemed to be natural and convenient conditions to get simple and applicable formulas. Their counterparts for non-isotropic random sets and particle processes are necessarily more complicated, as we have seen in some of the previous sections. However, once the step from isotropic to non-isotropic structures is made, the question arises whether a similar generalization from stationary to non-stationary structures is possible. Although random sets and point processes without any invariance properties have been studied by many authors under different aspects, one might get the impression that, for example, the mean value formulas for Boolean models, which are at the heart of stochastic geometry, rely on the invariance of the model. Surprisingly, this is not the case. As the dissertation of Fallert [222] showed (see also [223]), specific intrinsic volumes for Boolean models with convex or polyconvex grains can be introduced without any invariance requirements, and the formulas obtained in Section 9.1 transfer to this situation in a suitably generalized form. Even more astonishing is the fact that these local mean value formulas for non-stationary Boolean models (and Poisson particle processes) make heavy use of the iterated formulas of translative integral geometry, as we have discussed in Section 6.4. Thus, although we do not require that the distributions of our random structures are invariant with respect to the translation group, the corresponding integral geometric setting still plays an essential role.
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© 2008 Springer-Verlag Berlin Heidelberg
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(2008). Non-stationary Models. In: Stochastic and Integral Geometry. Probability and Its Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78859-1_11
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DOI: https://doi.org/10.1007/978-3-540-78859-1_11
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