A semicontinuity theorem and its application to network source coding

For a general class of Gaussian network source coding problems for which no single-letter characterization of the rate-distortion region is known, it is shown that the rate- distortion region is inner semicontinuous with respect to the source distribution. This semicontinuity theorem is used to obtain two new results, one for the Gaussian many-help-one problem and one for the Gaussian robust distributed source coding problem.