Optimality of Gaussian in Enlarging HK Rate Region, and its Overlap with the Capacity Region of 2-users GIC

This article shows that the set of HK constraints correspond to projecting the intersection of two multiple access channels on its sup-spaces. A key property of HK constraints is that the private message of user 1 (or of user 2) is the last layer in superposition coding for the MAC formed at receiver 2 (or at receiver 1) and will be treated as noise in the decoding operations at receiver 2 (or at receiver 1). This property is used in this article to show that, in a HK rate region based on an additive Gaussian noise model, Gaussian distribution is optimum for enlarging the region. It is known that the HK rate region is achievable in an interference channel. On the other hand, reference [Khandani-arXiv:2011.12981] presents a method for code-book construction in a 2-users weak GIC, using Gaussian inputs, based on covering the boundary of the capacity region in infinitesimal steps. The region constructed in [Khandani-arXiv:2011.12981] coincides with the intersection of the same two multiple access channels that surface in the HK rate region. Reference [Khandani-arXiv:2011.12981] also shows that, due to the aforementioned key property, 2-users GIC capacity region cannot extend beyond the HK rate region. This means, using Gaussian code-books, the HK rate region is optimally enlarged and the same Gaussian code-books achieve the capacity region of the 2-users GIC.