even to conjecture the inequality (\[eq:1\]). In fact the proof of the positive mass theorem turns out to be very subtle [@Schoen79b][@Schoen81][@witten81].

A key assumption in the positive mass theorem is that the matter fields should satisfy an energy condition. This condition is expected to hold for all physically realistic matter. This kind of general properties which do not depend very much on the details of the model are not easy to find for a macroscopic object. And hence it is difficult to obtain simple and general geometric inequalities among the parameters that characterize ordinary macroscopic objects. Black holes represent a unique class of very simple macroscopic objects and hence they are natural candidates for geometrical inequalities. Nevertheless, in section \[sec:open-problems-recent\] we will present also a geometric inequality valid for ordinary bodies.

The black hole uniqueness theorem ensures that stationary black holes in vacuum are characterized by the Kerr exact solution of Einstein equations [^1]. For simplicity we will not consider the electromagnetic field in this article, however most of the results presented here can be generalized to include that case.

It is somehow remarkable that the same family of solutions of Einstein equations that describe the unique stationary black 