  Nonconvex optimization plays a key role in multi-user information theory and
related fields, but it is usually difficult to solve. The rate region of the
Gray--Wyner source coding system (or almost equivalently, the mutual
information region) is a typical example in nonconvex optimization, whose
single-letter expression was given by Gray and Wyner. However, due to the
nonconvexity of the optimization involved in this expression, previously, there
was none nontrivial discrete source for which the analytic expression is known.
In this paper, we propose a new strategy to solve nonconvex optimization
problems. By this strategy, we provide the analytic expression for the doubly
symmetric binary source (DSBS), which confirms positively a conjecture of Gray
and Wyner in 1974. We also provide the analytic expression of the mutual
information region for the Gaussian source, and provide (or recover) the
analytic expressions of the lossy Gray--Wyner region for both the DSBS and
Gaussian source. Our proof strategy relies on an auxiliary measure technique
and the analytical expression of the optimal-transport divergence region.
