Entropy jumps in the presence of a spectral gap

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The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems (2004, background): “In fact, for the study of the function g defined in (3), one can apply the “Basic formula” of [2], which was extended to the d-dimensional case in [1] (although not in the form (13) reproduced here).”
On the rate of convergence in the entropic central limit theorem (2004, methodology): “The main result in [3] asserts that independent copies X1, X2 of such a random variable X satisfy Ent ( X1 + X2√ 2 ) − Ent(X) ≥ c 2 + 2c (Ent(G)− Ent(X)).”
Semi Log-Concave Markov Diffusions (2013, background): “The first result for CP is proved in [6] proposition 1.”
Solution of Shannon's problem on the monotonicity of entropy (2004, methodology): “A version of this theorem (in the case n = 2) was introduced in [2] and was motivated by an analysis of the transportation of measures and a local analogue of the Brunn-Minkowski inequality.”
Entropy jumps for isotropic log-concave random vectors and spectral gap (2012, methodology): “As mentioned above, we need to develop a method different from the one of [3].”