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.

中文翻译：

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DSBS最小相对熵区域包络的凸度和凹度

在本文中，我们证明了对于双对称二元分布，最小相对熵区域的下递增包络和上包络分别是凸的和凹的。我们还证明了另一个引起最小相对熵区域的函数是凹的。这两个包络和这个函数以前被用来表征强小集展开问题和强 Brascamp--Lieb 不等式中的最优指数。本文的结果，结合 Yu, Anantharam, and Chen (2021) 推导出的强小集展开定理，以及 Yu (2021) 推导出的强 Brascamp--Lieb 不等式，正面证实了 Ordentlich--Polyanskiy-- Shayevitz 关于强小集展开的猜想 (2019) 和 Polyanskiy 关于强 Brascamp--Lieb 不等式的猜想 (2016)。