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This paper attempts to look at the interconnections existing between metrics, convexity, and integral geometry from the point of view of combinatorial integral geometry. Along with general expository material, some new concepts and results are presented, in particular the sin2-representations of...
Stochastic calculus can be used to provide a satisfactory theory of random processes on differentiable manifolds and, in particular, a description of Brownian motion on a Riemannian manifold which lends itself to constructions generalizing the classical development of smooth paths on a manifold....
In this paper some isoperimetric inequalities for stationary random tessellations are discussed. At first, classical results on deterministic tessellations in the Euclidean plane are extended to the case of random tessellations. An isoperimetric inequality for the random Poisson polygon is...
This paper considers line processes and random mosaics. The processes are assumed invariant with respect to the group of translations ofR
2. An expression for the probabilities π,k=0, 1, 2,... to havek hits on an interval of lengtht taken on a ‘typical line of direction α’ (the hits are produced...
This paper gives a survey of the theory of point-set randomizations with respect to a group. It is shown that only the so-calledk-lattices are randomizable with respect to the parallel translations and all motions of theR
n space. It is shown how the randomizability problems are connected with...
Using the expectation of a particular random set, we present a multivariate extension of a characterization theorem involving extreme order statistics.
Point processesX of cylinders, compact sets (particles), or flats inR
are mathematical models for fields of sets as they occur, e.g., in practical problems of image analysis and stereology. For the estimation of geometric quantities of such fields, mean value formulas forX are important. By a...
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