On the Poincaré Constant of Log-Concave Measures

Functional inequalities for perturbed measures with applications to log-concave measures and to some Bayesian problems

“…This study was based on the fact that weak forms of the Poincaré inequality imply the usual form. A similar result was first stated by E. Milman ([34]) and the section 9.2 in [19] is devoted to extensions of E. Milman's results. Actually, in subsection 9.3.1 of [19] devoted to the transference via absolute continuity we missed the point.…”

“…The final result (3) is the most general one obtained by transference in the log-concave situation. [9] contains a lot of other results in this direction, [19] contains alternative, simpler but worse results. Notice that (3) is wrongly recalled in [19] where a square is missing.…”

“…[9] contains a lot of other results in this direction, [19] contains alternative, simpler but worse results. Notice that (3) is wrongly recalled in [19] where a square is missing. The remarkable property of (3) or (2) is that the bound is dimension free (but actually dimension is hidden either in the Lipschitz constant or in the choice of K in most of the generic examples).…”

“…In [19] we have studied several properties of the Poincaré constant for log-concave measures in particular the transference of these inequalities using absolute continuity, distance, mollification. This study was based on the fact that weak forms of the Poincaré inequality imply the usual form.…”

“…Since, a lot of works have been devoted to this conjecture, satisfied in some special cases. We refer to the book [2] for references before 2015, and to [19] for more information on the Poincaré constant of log-concave measures. Up to very recently the best general known result was…”