  We prove a warped product splitting theorem for manifolds with Ricci
curvature bounded from below in the spirit of [Croke-Kleiner, Duke Math. J.
(1992)], but instead of asking that one boundary component is compact and mean
convex, we require that it is parabolic and convex. The parabolicity assumption
cannot be dropped as, otherwise, the catenoid in ambient dimension four would
give a counterexample. As an application, we deduce a half-space theorem for
mean convex sets in product manifolds (resp. for sets whose boundary has mean
curvature bounded below by a definite constant, in warped products with
negative curvature). The results are obtained by combining glueing techniques
for manifolds and optimal transport tools from synthetic Ricci curvature
bounds.
