Interaction strictly improves the Wyner-Ziv rate-distortion function

In 1985 Kaspi provided a single-letter characterization of the sum-rate-distortion function for a two-way lossy source coding problem in which two terminals send multiple messages back and forth with the goal of reproducing each other´s sources. Yet, the question remained whether more messages can strictly improve the sum-rate-distortion function. Viewing the sum-rate as a functional of the distortions and the joint source distribution and leveraging its convex-geometric properties, we construct an example which shows that two messages can strictly improve the one-message (Wyner-Ziv) rate-distortion function. The example also shows that the ratio of the one-message rate to the two-message sum-rate can be arbitrarily large and simultaneously the ratio of the backward rate to the forward rate in the two-message sum-rate can be arbitrarily small.