On Linear Transformations of Intersections

Abstract

For any linear transformation and two convex closed sets, we provide necessary and sufficient conditions for when the transformation of the intersection of the sets coincides with the intersection of their images. We also identify analogous conditions for non-convex sets, general transformations, and multiple sets. We demonstrate the usefulness of our results via an application to the economics literature of mechanism design.

Keywords: linear transformation, convex closed set, intersection, directional convexity, mechanism design, dominant-strategy implementation, Bayesian implementation

JEL Classification: D82