  We consider the problem of source coding subject to a fidelity criterion for
the Gray-Wyner network that connects a single source with two receivers via a
common channel and two private channels. The pareto-optimal trade-offs between
the sum-rate of the private channels and the rate of the common channel is
completely characterized for jointly Gaussian sources subject to the
mean-squared error criterion, leveraging convex duality and an argument
involving the factorization of convex envelopes. Specifically, it is attained
by selecting the auxiliary random variable to be jointly Gaussian with the
sources.
