No abstract

We construct valuations on the space of finite-valued convex functions using integration of differential forms over the differential cycle associated to a convex function. We describe the kernel of this procedure and show, that the intersection of this space of smooth valuations with the space of all dually epi-translation invariant valuations on convex functions is dense in the latter. We also show that the same holds true for the space spanned by mixed Hessian valuations. This is a version of McMullen's conjecture for valuations on convex functions.

We study continuous, dually epi-translation invariant valuations on C n that are invariant under the unitary group and we give a description of all valuations belonging to the dense subspace of smooth valuations. In addition, we give a classification of all simple valuations and we introduce a version of the Klain function for valuations of this type. These results are then used to show that a unitarily invariant valuation is uniquely determined by its restriction to a certain finite family of subspaces of C n .

We give a new proof of the Hadwiger theorem on convex functions derived from a characterization of smooth rotation invariant valuations. We also give a description of the construction of singular Hessian valuations using integration over the differential cycle and provide a new representation of these functionals as principal value integrals with respect to the Hessian measures.

We show how the classification of continuous, epi-translation invariant valuations on convex functions of maximal degree of homogeneity established by Colesanti, Ludwig, and Mussnig can be obtained from a classical result of McMullen by explicitly relating these functionals to valuations on higher dimensional convex bodies. Following this geometric interpretation, we derive a new description of 1-homogeneous, continuous, and epi-translation invariant valuations on convex functions analogous to a classical result by Goodey and Weil.