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Determination of the mean normal measure from isotropic means of flat sections
- Part of
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- Journal:
- Advances in Applied Probability / Volume 34 / Issue 3 / September 2002
- Published online by Cambridge University Press:
- 01 July 2016, pp. 505-519
- Print publication:
- September 2002
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- Article
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Let
be the mean normal measure of a stationary random set Z in the extended convex ring in ℝd. For k ∈ {1,…,d-1}, connections are shown between 
and the mean of 
. Here, the mean is understood to be with respect to the random isotropic k-dimensional linear subspace ξk and the mean normal measure of the intersection is computed in ξk. This mean to be well defined, a suitable spherical lifting must be applied to 
before averaging. A large class of liftings and their resulting means are discussed. In particular, a geometrically motivated lifting is presented, for which the mean of liftings of 
determines 
uniquely for any fixed k ∈ {2,…,d-1}.
be the mean normal measure of a stationary random set 
and the mean of 
. Here, the mean is understood to be with respect to the random isotropic 
before averaging. A large class of liftings and their resulting means are discussed. In particular, a geometrically motivated lifting is presented, for which the mean of liftings of 
determines 
uniquely for any fixed 