Consider a stationary Boolean model X with convex grains in Rd and let any exposed lower tangent point of X be shifted towards the hyperplane N0={x∈Rd:x1=0} by the length of the part of the segment between the point and its projection onto the N0 covered by X. The resulting point process in the halfspace (the Laslett's transform of X) is known to be stationary Poisson and of the same intensity as the original Boolean model. This result was first formulated for the planar Boolean model (see N. Cressie \cite{Cressie}) although the proof based on discretization is partly heuristic and not complete. Starting from the same idea we present a rigorous proof in the d-dimensional case. As a technical tool equivalent characterization of vague convergence for locally finite integer valued measures is formulated. Another proof based on the martingale approach was presented by A. D. Barbour and V. Schmidt \cite{barb+schm}.
A new method of testing the random closed set model hypothesis (for example: the Boolean model hypothesis) for a stationary random closed set Ξ⊆\rd with values in the extended convex ring is introduced. The method is based on the summary statistics - normalized intrinsic volumes densities of the \ep-parallel sets to Ξ. The estimated summary statistics are compared with theirs envelopes produced from simulations of the model given by the tested hypothesis. The p-level of the test is then computed via approximation of the summary statistics by multinormal distribution which mean and the correlation matrix is computed via given simulations. A new estimator of the intrinsic volumes densities from \cite{MR06} is used, which is especially suitable for estimation of the intrinsic volumes densities of \ep-parallel sets. The power of this test is estimated for planar Boolean model hypothesis and two different alternatives and the resulted powers are compared to the powers of known Boolean model tests. The method is applied on the real data set of a heather incidence.