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 th intrinsic volume of a compact convex set in can be obtained as an invariant integral of the th intrinsic volume of sections with k-planes. This paper discusses the question if the 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 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 or when and we restrict considerations to even measurement functions. Additional examples of local functionals can be constructed when .