Simple valuations on convex bodies

A valuation on the space H-d of convex bodies in Euclidean d-space R(d) is a real function phi on H-d that satisfies phi(H boolean OR L) + phi(K boolean AND L) = phi(K) + phi(L) whenever K, L, K boolean OR L epsilon H-d (thus, only real valued valuations are considered in this note). The relevance of valuations for the theory of convex bodies can be seen from the surveys given by McMullen and Schneider [7] and by McMullen [6]. For notions related to convex bodies that will be used in the following, we refer to [11]. An important theorem of Hadwiger [2] characterizes the continuous rigid motion invariant valuations on H-d as the linear combinations of intrinsic volumes. A remarkable new and simpler proof of this result was recently given by Klain [3]. It would be interesting to have a counterpart to Hadwiger's characterization theorem with rigid motion invariance replaced by translation invariance. As a byproduct of his new proof, Klain obtained the following characterization of the volume V-d. A valuation on H-d is called simple if it is zero on bodies of dimension less than d.