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Uniqueness of the measurement function in Crofton's formula

By Rikke Eriksen and Markus Kiderlen
CSGB Research Reports
No. 07, August 2019

Crofton's intersection formula states that the $(n-j)$'th intrinsic volume of a compact convex set in $\mathbb{R}^n$ 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 $k \geq 2$.

Keywords: Crofton's formula, Klain functional, Local functions, Spherical lifting, Uniqueness, Valuation

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