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The Convexity and Concavity of Envelopes of the Minimum-Relative-Entropy Region for the DSBS

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arxiv 2106.03654 v1 pith:FH3OTEI3 submitted 2021-06-07 cs.IT math.ITmath.PR

Lei Yu This is my paper · ORCID
classification cs.ITmath.ITmath.PR
keywords strongbrascamp--liebconvexityexpansionfunctionminimum-relative-entropyregionsmall-set
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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.

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Cited by 2 Pith papers

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