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An application of Crofton’s formula to moment measures of random curvature measures. (English) Zbl 0618.60019
Forschungsergeb., Friedrich-Schiller-Univ. Jena N/87/12, 1-12 (1987).
Consider a random closed set in the d-dimensional Euclidean space \({\mathbb{R}}^ d\) for which all curvature measures are defined (with prob. 1) and become random measures. The higher moment measures of these random curvature measures are expressed by section measures with planes by the help of Crofton’s formulae. A special formula is derived for the case of stationary random sets.
A detailed discussion of the situation in \({\mathbb{R}}^ 2\) and \({\mathbb{R}}^ 3\) leads to a known formula which expresses the second moment volume measure as a weighted integral of the set covariance but also to an apparently new formula for the second moment of the boundary length for a planar set.

MSC:
60D05 Geometric probability and stochastic geometry
60G57 Random measures