  We generalize McCann's theorem of optimal transport to a submanifold setting
and prove Michael-Simon-Sobolev inequalities for submanifolds in manifolds with
lower bounds on intermediate Ricci curvatures. The results include a variant of
the sharp Michael-Simon-Sobolev inequality in S. Brendle's work
arXiv:2009.13717 when the intermediate Ricci curvatures are nonnegative.
