  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.
