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Rotation Invariant Valuations

By Eva B.Vedel Jensen and Markus Kiderlen
CSGB Research Reports
No. 03, February 2016
This paper contains an overview of the results available in the literature, concerning characterization of rotation invariant valuations. In particular, we discuss the characterization theorem, derived in [1], for continuous rotation invariant polynomial valuations on the set $\mathcal{K}^n$ of convex bodies in $\mathbb{R}^n$. Next, rotational Crofton formulae are presented. Using new kinematic formulae for trace-free tensor valuations, it is possible to extend the rotational Crofton formulae for tensor valuations, available in the literature. Principal rotational formulae for tensor valuations are also discussed. These formulae can be derived using locally defined tensor valuations. A number of open questions in rotational integral geometry {are} presented.
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