Title: A Counterexample to the Vector Generalization of Costa's EPI, and Partial Resolution,
Abstract: We give a counterexample to the vector generalization of Costa's entropy
power inequality (EPI) due to Liu, Liu, Poor and Shamai. In particular, the
claimed inequality can fail if the matix-valued parameter in the convex
combination does not commute with the covariance of the additive Gaussian
noise. Conversely, the inequality holds if these two matrices commute.