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.
Valuations on Log-Concave Functions
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abstract
A classification of $\operatorname{SL}(n)$ and translation covariant Minkowski valuations on log-concave functions is established. The moment vector and the recently introduced level set body of log-concave functions are characterized. Furthermore, analogs of the Euler characteristic and volume are characterized as $\operatorname{SL}(n)$ and translation invariant valuations on log-concave functions.
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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.