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.