Continuous Rotation Invariant Valuations on Convex Sets

The notion of valuation on convex sets can be considered as a generalization of the notion of measure, which is deflned only on the class of convex compact sets. It is well-known that there are important and interesting examples of valuations on convex sets, which are not measures in the usual sense as, for example, the mixed volumes. Basic deflnitions and some classical examples are discussed in Section 2 of this paper. For more detailed information we refer to the surveys [Mc-Sch] and [Mc3]. Throughout this paper all the valuations are assumed to be continuous with respect to the Hausdorfi metric. Note that the theory of valuations which are invariant or covariant with respect to translations belongs to the classical part of convex geometry. There exists an explicit description of translation invariant continuous valuations on 1 and 2 due to Hadwiger [H1] (the case of 2 is nontrivial). Continuous rigid motion invariant valuations on d are completely classifled by the remarkable Hadwiger theorem as linear combinations of the quermassintegrals (cf. [H2] or for a simpler proof [K]).