theory. It is often the case that the quantities involved have a clear physical interpretation and the expected behavior of the gravitational and matter fields often suggests geometric inequalities which can be highly non-trivial from the mathematical point of view. The interplay between physics and geometry gives to geometric inequalities in General Relativity their distinguished character. These inequalities relate quantities that have both a physical interpretation and a geometrical definition.

The plan of this article follows this interplay between physics and mathematics. In section \[sec:physical-picture\] we present the physical motivations for the black holes geometric inequalities. In section \[sec:theorems\] we summarize some theorems where these inequalities have been recently proved. Finally, in section \[sec:open-problems-recent\] we list relevant open problems and we also describe recent results on geometric inequalities for bodies.

Physical picture {#sec:physical-picture}
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An important example of a geometric inequality is the positive mass theorem. Let $m$ be the total ADM mass on an asymptotically flat complete initial data such that the dominant energy condition is satisfied. Then we have $$\label{eq:1}
0\leq m\quad  (=\text{ Minkowski}).$$ The mass $m$ is a pure geometrical quantity [@Arnowitt62][@Bartnik86][@chrusciel86]. However, from the geometrical mass definition, without the physical picture, it would be very hard 