Convex valuations theory lies between integral and convex geometry. A continuous convex valuation is just a finitely additive measure on convex bodies, which depends on the body continuously. Some classical results that fit naturally into the domain of convex valuations include Crofton's formula for the length of a curve, Chern's kinematic formulas, and Hadwiger's characterization theorem. The latter identifies all the continuous valuations that are invariant under rigid motions as the quermassintegrals.
What happens if the Euclidean group of motions is replaced by other groups? Following Alesker's proof of McMullen's conjecture fifteen years ago, a lot of algebraic structure has been uncovered on the space of valuations. Those developments allowed to study the integral geometry of other groups, such as the unitary and the symplectic groups. For most interesting compact groups, Hadwiger-type classification results, as well as general structure theorems on the spaces of invariant valuations, have been obtained by Alesker, Bernig, Fu, Solanes and others. However, for non-compact groups, a lot of open questions remain. For instance, the integral-geometric notion appearing naturally is the generalized valuation, for which less structure is available. In the talk, I will describe some structure on the generalized valuations, and classification results in the presence of certain non-compact groups such as the Lorentz group, obtained in joint works with Alesker and Bernig.