A Hadwiger-Type Theorem for General Tensor Valuations

Abstract

Hadwiger’s characterization of continuous rigid motion invariant real valued valuations has been the starting point for many important developments in valuation theory. In this chapter, the decomposition of the space of continuous and translation invariant valuations into a sum of SO(n) irreducible subspaces, derived by S. Alesker, A. Bernig and the author, is discussed. It is also explained how this result can be reformulated in terms of a Hadwiger-type theorem for translation invariant and SO(n) equivariant valuations with values in an arbitrary finite dimensional SO(n) module. In particular, this includes valuations with values in general tensor spaces. The proofs of these results will be outlined modulo a couple of basic facts from representation theory. In the final part, we survey a number of special cases and applications of the main results in different contexts of convex and integral geometry.