Convex Order and Moment Persuasion

If a probability measure is a mean-preserving contraction of another, the two measures are said to be in convex order. This paper provides a complete characterization of mean-preserving contractions and their extreme points in the multidimensional setting, extending the results of Kleiner, Moldovanu, Strack, and Whitmeyer (2024) to incorporate fully revealing regions and lower-dimensional pooling regions. A central feature of our characterization is a "concentration " structure: each state can be mapped only to posterior means lying within its own irreducible component. We apply these results to the moment persuasion problem.