  In this paper, we prove that for the doubly symmetric binary distribution,
the lower increasing envelope and the upper envelope of the
minimum-relative-entropy region are respectively convex and concave. We also
prove that another function induced the minimum-relative-entropy region is
concave. These two envelopes and this function were previously used to
characterize the optimal exponents in strong small-set expansion problems and
strong Brascamp--Lieb inequalities. The results in this paper, combined with
the strong small-set expansion theorem derived by Yu, Anantharam, and Chen
(2021), and the strong Brascamp--Lieb inequality derived by Yu (2021), confirm
positively Ordentlich--Polyanskiy--Shayevitz's conjecture on the strong
small-set expansion (2019) and Polyanskiy's conjecture on the strong
Brascamp--Lieb inequality (2016). The proofs in this paper are based on the
equivalence between the convexity of a function and the convexity of the set of
minimizers of its Lagrangian dual.
