We study stochastic properties of the empty space for stationary germ-grain models in Rd; in particular, we deal with the inner radius of the empty space with respect to a general structuring element which is allowed to be lower dimensional. We consider Poisson cluster and mixed Poisson germ-grain models, and show in several situations that more variability results in stochastically greater empty space in terms of the empty space hazard function. Furthermore, we study the asymptotic behaviour of the empty space hazard functions at 0 and at ∞.

A configurational method for describing shape is proposed based on two measures that gauge human experiences of moving through space: distance and changes in direction of travel. Boundary shapes from the built environment and nature are studied in a morphospace composed of two axes: one corresponding to each measure, to yield a typological classification of form. It is shown that the covariance between distance and directionality is mediated by the topological structure of embedded main circulation. Three kinds of circulation—elementary, ring, and linear—thus affect three fundamentally different balancing conditions between distance and directionality in boundary shapes. The analysis of large samples of shapes thus far demonstrates a “unique shape” status, where no two different shapes have the same pair of relative distance and directional fragmentation values.