Before revision: As a byproduct, we derive a number of applications to law-invariant acceptance sets and risk measures as well as Schur-convex functionals .
After revision: As a byproduct, we derive a number of applications to law-invariant functionals: the equivalence of law invariance and Schur convexity, i.e., monotonicity with respect to the convex stochastic order, and the fact that a law-invariant functional is fully determined by its behaviour on bounded random variables. We show how to apply these results to provide a unifying perspective on the literature on law-invariant functionals, with special emphasis on quantile-based representations, including Kusuoka representations, dilatation monotonicity, and infimal convolutions .