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Uniqueness of the measurement function in Crofton's formula
Advances in Applied Mathematics ( IF 0.952 ) Pub Date : 2020-01-31 , DOI: 10.1016/j.aam.2020.102004
Rikke Eriksen, Markus Kiderlen

Crofton's intersection formula states that the (nj)th intrinsic volume of a compact convex set in Rn can be obtained as an invariant integral of the (kj)th intrinsic volume of sections with k-planes. This paper discusses the question if the (kj)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 (kj)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 k2.

更新日期:2020-01-31

 

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