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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 \cite{fkm}, 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 products, 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.