Random cell complexes and generalised sets
The concepts of random (curved) cell complexes, generalised sets, and their mean sets in \(R^ d\) are introduced. Under stationarity conditions various relations between associated geometrical quantities are derived. A central equation is the following stochastic mean value version of an extension of the Euler polyhedron theorem \((k=0)\) to Lipschitz-Killing curvature measures: \[ c^ i_ k =\sum^{i}_{j=k}(-1)^{j-k} N^ jC^ j_ k,\quad 0\leq k\leq i\leq d-1. \] Here \(c^ i_ k\) denotes the k-th Lipschitz-Killing curvature density of the random i-skeleton, \(N^ j\) the mean number of j-cells per unit volume, and \(C^ j_ k\) the mean total k-th Lipschitz-Killing curvature of the typical j-cell. (The surface area densities \(c^ i_ i\) and the mean surface areas \(C^ j_ j\) are contained as margins.) In the case of random tessellations the formulas may be further specified and the known results for convex mosaics in \(R^ 2\), \(R^ 3\) are extended.
scientific article; zbMATH DE number 4072017