Theoretical study demonstrates the classification of invariant valuations on convex Lipschitz functions, highlighting structural foundations of functional affine surface areas.
A characterization of valuations on the space of convex Lipschitz functions whose domain is a polytope in Rⁿ is obtained. It is shown that every upper semicontinuous, equi-affine and dually epi-translation invariant valuation can be written as a linear combination of a constant term, the volume of the domain, and a functional affine surface area. In addition, dual statements for finite-valued convex functions are established.
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