A new dual to the Gács-Körner common information defined via the Gray-Wyner system

We consider jointly distributed random variables X and Y. After describing the Gács-Körner common information between the random variables from the viewpoint of the capacity region of the Gray-Wyner system, we propose a new notion of common information between the random variables that is dual to the Gács-Körner common information from this viewpoint in a well-defined sense. We characterize this quantity explicitly in terms of two auxiliary quantities that are asymmetric in nature, and illustrate the operational significance of these new quantities by characterizing a corner point of the solution to a problem of source coding with side-information in terms of them. We also contrast this new concept of common information for a pair of random variables with the Wyner common information of the random variables, which is also a kind of dual to the Gács-Körner common information.