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.
$\operatorname{SL}(n)$ invariant valuations on super-coercive convex functions
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abstract
All non-negative, continuous, $\operatorname{SL}(n)$ and translation invariant valuations on the space of super-coercive, convex functions on $\mathbb{R}^n$ are classified. Furthermore, using the invariance of the function space under the Legendre transform, a classification of non-negative, continuous, $\operatorname{SL}(n)$ and dually translation invariant valuations is obtained. In both cases, different functional analogs of the Euler characteristic, volume and polar volume are characterized.
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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.