Rotation Invariant Valuations

In this chapter, we focus on rotation invariant valuations. We give an overview of the results available in the literature, concerning characterization of such valuations. In particular, we discuss the characterization theorem, derived in Alesker (Ann Math 149: 977-1005, 1999), for continuous rotation invariant polynomial valuations on K-n. Next, rotational Crofton formulae are presented. Using the new kinematic formulae for trace-free tensor valuations presented in Chap. 4, 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, as introduced in Chap. 2. A number of open questions in rotational integral geometry are presented.