  Wyner's Common Information and a natural relaxation are studied in the
special case of Gaussian random variables. The relaxation replaces conditional
independence by a bound on the conditional mutual information. The main
contribution is the proof that Gaussian auxiliaries are optimal, leading to a
closed-form formula. As a corollary, the proof technique also establishes the
optimality of Gaussian auxiliaries for the Gaussian Gray-Wyner network, a
long-standing open problem.
