From valuations on convex bodies to convex functions (Q6634481)

A real-valued lattice valuation is an order preserving real modular function. The authors produce valuations on the space of compactly supported super-coersive convex functions on \({\mathbb R}^n\); cf. [\textit{A. Colesanti} et al., Int. Math. Res. Not. 2019, No. 8, 2384--2410 (2019; Zbl 1436.52014)].\N\NA valuation of a function under consideration is defined as a valuation of the lower part of its epigraph. The authors study the properties of this tranformation and describe continuous epi-translation invariant valuations in the wake of \textit{P. McMullen} [Arch. Math. 34, 377--384 (1980; Zbl 0424.52003)] and \textit{P. Goodey} and \textit{W. Weil} [Proc. Lond. Math. Soc. (3) 49, 504--516 (1984; Zbl 0526.52004)].