Valuations on Convex Functions

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All continuous, SL$(n)$ and translation invariant valuations on the space of convex functions on ${\mathbb R}^n$ are completely classified.

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Valuations on convex functions and convex sets and Monge–Ampère operators (2017, background): “ion of valuation have been suggested. One of these recent generalizations is the notion of valuation on a class of functions introduced and studied by M. Ludwig and others, see e.g. [14], [15], [16], [10], [21], [22]. Let us give a deﬁnition of valuation on convex functions which is relevant for the”
A homogeneous decomposition theorem for valuations on convex functions (2019, unknown): “Here classification results were obtained for SL(n) invariant and for monotone valuations in [8, 14, 15, 32–34] and the connection to valuations on convex bodies was explored by Alesker [3].”
The Hadwiger Theorem on Convex Functions, I (2020, result): “Here classification results were obtained for SL(n) invariant valuations in [12, 13, 37, 38], the connection to valuations on convex bodies was explored in [3] and first structural results were established in [14,27,28].”