  In decentralized control problems, a standard approach is to specify the set
of allowable decentralized controllers as a closed subspace of linear
operators. This then induces a corresponding set of Youla parameters. Previous
work has shown that quadratic invariance of the controller set implies that the
set of Youla parameters is convex. In this paper, we prove the converse. We
thereby show that the only decentralized control problems for which the set of
Youla parameters is convex are those which are quadratically invariant. We
further show that under additional assumptions, quadratic invariance is
necessary and sufficient for the set of achievable closed-loop maps to be
convex. We give two versions of our results. The first applies to bounded
linear operators on a Banach space and the second applies to (possibly
unstable) causal LTI systems in discrete or continuous time.
