  We study a Riemannian manifold equipped with a density which satisfies the
Bakry--\'Emery Curvature-Dimension condition (combining a lower bound on its
generalized Ricci curvature and an upper bound on its generalized dimension).
We first obtain a Poincar\'e-type inequality on its boundary assuming that the
latter is locally-convex; this generalizes a purely Euclidean inequality of
Colesanti, originally derived as an infinitesimal form of the Brunn-Minkowski
inequality, thereby precluding any extensions beyond the Euclidean setting. A
dual version for generalized mean-convex boundaries is also obtained, yielding
spectral-gap estimates for the weighted Laplacian on the boundary. Motivated by
these inequalities, a new geometric evolution equation is proposed, which
extends to the Riemannian setting the Minkowski addition operation of convex
domains, a notion thus far confined to the purely linear setting. This
geometric flow is characterized by having parallel normals (of varying
velocity) to the evolving hypersurface along the trajectory, and is intimately
related to a homogeneous Monge-Amp\`ere equation on the exterior of the convex
domain. Using the aforementioned Poincar\'e-type inequality on the boundary of
the evolving hypersurface, we obtain a novel Brunn--Minkowski inequality in the
weighted-Riemannian setting, amounting to a certain concavity property for the
weighted-volume of the evolving enclosed domain. All of these results appear to
be new even in the classical non-weighted Riemannian setting.
