Constraint equations, Mass-momentum inequalities. Some of the most well-known aspects of the mathematical study of Einstein's equations are the proofs of positivity of the ADM mass for isolated systems. This has raised the challenge of generalizing and strengthening the control one has on the mass to the setting of black holes. An example would be the stipulated Penrose inequality which asserts that the size of black holes should provide a lower bound on the mass. Originally proposed by Penrose as a form of evidence in favor of his proposed weak cosmic censorship conjecture and what he termed the "establishment view" on the evolution and nal state of dynamical black holes, this inequality and its generalizations (including angular momentum) has attracted much attention, with a resolution in the time-symmetric case about thirteen years ago. We feel that recent progress on the dynamical black holes might be useful in these questions. Such proposed inequalities have been studied in conjunction with the constraint equations for space-like initial data sets. This topic is often also studied numerically, due to its usefulness of numerical simulations to the understanding of the dynamical evolution of the Einstein equations. Non-uniqueness in certain formulations of the Einstein constraint equations was rst discovered numerically. Moreover, numerical evolutions of black holes regularly monitor the black holes for violations of the bound of black hole spin on a Kerr black hole, as a possible indication of violation of cosmic censorship.