  We prove splitting theorems for mean convex open subsets in RCD (Riemannian
curvature-dimension) spaces that extend results by Kasue, Croke and Kleiner for
Riemannian manifolds with boundary to a non-smooth setting. A corollary is for
instance Frankel's theorem. Then, we prove that our notion of mean curvature
bounded from below for the boundary of an open subset is stable w.r.t. to
uniform convergence of the corresponding boundary distance function. We apply
this to prove almost rigidity theorems for uniform domains whose boundary has a
lower mean curvature bound.
