Representation of Quasi-Monotone Functionals by Families of Separating Hyperplanes

We characterize when the level sets of a continuous quasi-monotone functional defined on a suitable convex subset of a normed space can be uniquely represented by a family of bounded continuous functionals. Furthermore, we investigate how regularly these functionals depend on the parameterizing level. Finally, we show how this question relates to the recent problem of property elicitation that simultaneously attracted interest in machine learning, statistical evaluation of forecasts, and finance.