  In this paper, we consider a dilation type inequality on a weighted
Riemannian manifold, which is classically known as Borell's lemma in
high-dimensional convex geometry. We investigate the dilation type inequality
as an isoperimetric type inequality by introducing the dilation profile and
estimate it by the one for the corresponding model space under lower weighted
Ricci curvature bounds. We also explore functional inequalities derived from
the comparison of the dilation profiles under the nonnegative weighted Ricci
curvature. In particular, we show several functional inequalities related to
various entropies.
