Increasing functions and inverse Santaló inequality for unconditional functions

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Some functional inverse Santalo inequalities (2008, background): “Equivalently, the result of [8] implies that there exists c > 0 such that P(f ) ! c.”
Entropy jumps for isotropic log-concave random vectors and spectral gap (2012, background): “2) − log f(0) ≤ Ent(X) ≤ − log f(0) + d is for instance implicit in [14] and the easy proof is as follows.”
Functional Versions of Lp-Affine Surface Area and Entropy Inequalities (2014, unknown): “For all λ > 1, we go back to Corollary 1 and deduce that By the asymptotic functional reverse Santal´o inequality [21] (see also [27] in the even case), there exists a constant c > 0 such that R n e − ψ R n e − ψ ∗ ≥ c n .”
The volume product of convex bodies with many hyperplane symmetries (2013, methodology): “The strategy will be to divide K according to the cones generated by the facets of P and to adapt Meyer’s argument of [14] in this context.”