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 defined 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 definitions 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 Hausdorff 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 R1 and R2 due to Hadwiger [H1] (the case of R2 is nontrivial). Continuous rigid motion invariant valuations on Rd are completely classified by the remarkable Hadwiger theorem as linear combinations of the quermassintegrals (cf. [H2] or for a simpler proof [K]). There are two natural ways to generalize Hadwiger’s theorem: the first one is to describe continuous translation invariant valuations without any assumption on rotations; the second one is to characterize continuous rotation (i.e. either O(d)or SO(d)-) invariant valuations without any assumption on translations (here O(d) denotes the full orthogonal group and SO(d) denotes the special orthogonal group). The first problem is of interest to classical convexity and translative integral geometry. As we have said, it was solved by Hadwiger for the line and the 2-dimensional plane. There is a conjecture due to P. McMullen [Mc2], which states that every continuous translation invariant valuation can be approximated (in some sense) by linear combinations of mixed volumes (note that in the 3-dimensional space this conjecture is known to be true and it follows from several other general results, which we do not discuss here).