Ludwig, M., & Li, J. (2024, July 15). Valuations on Convex Functions [Conference Presentation]. 9th European Congress of Mathematics, Sevilla, Spain. http://hdl.handle.net/20.500.12708/199538
Let \(\operatorname{Conv}(\mathbb R^n)\) be the space of proper, lower semicontinuous, convex functions \(v\colon \mathbb R^n\to (-\infty,\infty]\) and \(\mathbb A\) an Abelian semi-group. A functional \(\operatorname{Z}\colon \operatorname{Conv}(\mathbb R^n)\to \mathbb A \) is called a {\em valuation} if
\[
\operatorname{ Z}(f\vee g)+\operatorname{Z}(f\wedge g)=\operatorname{Z}(f) +\operatorname{Z} (g)
\]
for all \(f,g\in \operatorname{Conv}(\mathbb R^n)\) such that the pointwise maximum \(f\vee g\) and the pointwise minimum \(f\wedge g\) are in \( \operatorname{Conv}(\mathbb R^n)\). We present classification results of real and measure-valued valuations on convex functions on \(\mathbb R^n\).
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Project title:
Bewertungen auf konvexen Funktionen: P 34446-N (FWF - Österr. Wissenschaftsfonds)