We obtain a new proof of Bobkov's lower bound on the first positive
eigenvalue of the (negative) Neumann Laplacian (or equivalently, the Cheeger
constant) on a bounded convex domain $K$ in Euclidean space. Our proof avoids
employing the localization method or any of its geometric extensions. Instead,
we deduce the lower bound by invoking a spectral transference principle for
log-concave measures, comparing the uniform measure on $K$ with an
appropriately scaled Gaussian measure which is conditioned on $K$.

A theorem of L. Caffarelli implies the existence of a map pushing forward a
source Gaussian measure to a target measure which is more log-concave than the
source one, which contracts Euclidean distance (in fact, Caffarelli showed that
the optimal-transport Brenier map $T_{opt}$ is a contraction in this case).

Various properties of isoperimetric, functional, Transport-Entropy and
concentration inequalities are studied on a Riemannian manifold equipped with a
measure, whose generalized Ricci curvature is bounded from below. First,
stability of these inequalities with respect to perturbation of the measure is
obtained. The extent of the perturbation is measured using several different
distances between perturbed and original measure, such as a one-sided
$L^\infty$ bound on the ratio between their densities, Wasserstein distances,
and Kullback - Leibler divergence.