  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.
