We prove an isoperimetric-type inequality for maximal, spacelike submanifold in the Minkowski space. The argument is based on the recent work of Brendle.

We generalize McCann's theorem of optimal transport to a submanifold setting and use it to 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 [arXiv:2009.13717] when the intermediate Ricci curvatures are nonnegative. 2020 Mathematics Subject Classification. Primary 53C21, 49Q22; Secondary 53C23.

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