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Uniqueness of the measurement function in Crofton's formula
Advances in Applied Mathematics ( IF 1.008 ) Pub Date : 2020-01-31 , DOI: 10.1016/j.aam.2020.102004
Rikke Eriksen; Markus Kiderlen

Crofton's intersection formula states that the (n−j)th intrinsic volume of a compact convex set in Rn can be obtained as an invariant integral of the (k−j)th intrinsic volume of sections with k-planes. This paper discusses the question if the (k−j)th intrinsic volume can be replaced by other functionals, that is, if the measurement function in Crofton's formula is unique. The answer is negative: we show that the sums of the (k−j)th intrinsic volume and certain translation invariant continuous valuations of homogeneity degree k yield counterexamples. If the measurement function is local, these turn out to be the only examples when k=1 or when k=2 and we restrict considerations to even measurement functions. Additional examples of local functionals can be constructed when k≥2.
更新日期:2020-01-31

 

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