A Klain-Schneider Theorem for Vector-Valued Valuations on Convex Functions

Abstract: A functional analog of the Klain-Schneider theorem for vector-valued valuations on convex functions is established, providing a classification of continuous, translation covariant, simple valuations. Under additional rotation equivariance assumptions, an analytic counterpart of the moment vector is characterized alongside a new epi-translation invariant valuation. The former arises as the top-degree operator in a family of functional intrinsic moments, which are linked to functional intrinsic volumes through translations. The latter represents the top-degree operator in a class of Minkowski vectors, which are introduced in this article and which lack classical counterparts on convex bodies, as they vanish due to the Minkowski relations. Additional classification results are obtained for homogeneous valuations of extremal degrees.