Convexity and multi-dimensional screening for spaces with different dimensions

We study the principal-agent problem. We show that b-convexity of the space of products, a condition which appears in a recent paper by Figalli, Kim and McCann (2011) [9], is necessary to formulate the problem as a maximization over a convex set. We then show that when the dimension m of the space of types is larger than the dimension n of the space of products, this condition implies that the extra dimensions do not encode independent economic information. When m is smaller than n, we show that under b-convexity of the space of types, it is always optimal for the principal to offer goods only from a certain prescribed subset. We show that this is equivalent to offering an m-dimensional space of goods.