Continuous, epi-translation invariant valuations on super-coercive convex functions decompose into homogeneous components of degrees 0 through n, and the degree-n valuations are exactly integrals of compactly supported functions of the gradient.
Alesker, Description of translation invariant valuations on convex sets with solution of P
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A homogeneous decomposition theorem for valuations on convex functions
Continuous, epi-translation invariant valuations on super-coercive convex functions decompose into homogeneous components of degrees 0 through n, and the degree-n valuations are exactly integrals of compactly supported functions of the gradient.