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We review some recent results about the relations among isoperimetric sets,Penrose inequalities and related concepts in the analysis of $3$-manifolds ofnonnegative scalar curvature. We also show that if the isoperimetric sets ofbig volume have connected boundaries, the equivalence among suitable notions ofmass hold.

Isoperimetric sets in nonnegative scalar curvature and their role through various concepts of mass