  The goal of this paper is to push forward the study of those properties of
log-concave measures that help to estimate their Poincar{\'e} constant. First
we revisit E. Milman's result [40] on the link between weak (Poincar{\'e} or
concentration) inequalities and Cheeger's inequality in the logconcave cases,
in particular extending localization ideas and a result of Latala, as well as
providing a simpler proof of the nice Poincar{\'e} (dimensional) bound in the
inconditional case. Then we prove alternative transference principle by
concentration or using various distances (total variation, Wasserstein). A
mollification procedure is also introduced enabling, in the logconcave case, to
reduce to the case of the Poincar{\'e} inequality for the mollified measure. We
finally complete the transference section by the comparison of various
probability metrics (Fortet-Mourier, bounded-Lipschitz,...).
