  We study the infimum of the best constant in a functional inequality, the
Brascamp-Lieb-like inequality, over auxiliary measures within a neighborhood of
a product distribution. In the finite alphabet and the Gaussian cases, such an
infimum converges to the best constant in a mutual information inequality.
Implications for strong converse properties of two common randomness (CR)
generation problems are discussed. In particular, we prove the strong converse
property of the rate region for the omniscient helper CR generation problem in
the discrete and the Gaussian cases. The latter case is perhaps the first
instance of a strong converse for a continuous source when the rate region
involves auxiliary random variables.
