The Extreme Points of Mean-Preserving Contractions

We study multidimensional mean-preserving contractions (MPC) and their extreme and exposed points. Proposition 1 focuses on extreme MPCs of a measure μ. Necessarily, each finitely supported extreme MPC ν induces a partition of X, the domain of μ, in convex sets such that the support of ν on each element of the partition is an affinely independent set, and such that the restriction of ν on each element of the partition is itself a MPC of the restriction of the prior μ on that element. Proposition 2 connects finitely supported Lipschitz-exposed points (measures that are unique optimizers of Lipschitz-continuous objectives) and power diagrams, which are divisions of a space into convex polyhedral cells according to a weighted proximity criterion. Power diagrams are very useful in a number of economic applications such as optimal transport and mechanism design. Finally, we apply the above results to several questions concerning moment persuasion and categorization