Geometrical inequalities show how certain parameters of a physical system set restrictions on other parameters. For instance, a black hole of given mass can not rotate too fast, or an ordinary object of given size can not have too much electric charge. In this article, we are interested in bounds on the angular momentum and electromagnetic charges, in terms of total mass and size. We are mainly concerned with inequalities for black holes and ordinary objects. The former are the most studied systems in this context in General Relativity, and where most results have been found. Ordinary objects, on the other hand, present numerous challenges and many basic questions concerning geometrical estimates for them are still unanswered. We present the many results in these areas. We make emphasis in identifying the mathematical conditions that lead to such estimates, both for black holes and ordinary objects.

几何不等式显示了物理系统的某些参数如何对其他参数设置限制。例如，给定质量的黑洞不能旋转太快，或者给定大小的普通物体不能有太多电荷。在本文中，我们感兴趣的是角动量和电磁电荷在总质量和尺寸方面的界限。我们主要关心黑洞和普通物体的不等式。前者是广义相对论中研究最多的系统，也是发现最多结果的系统。另一方面，普通物体提出了许多挑战，并且有关它们几何估计的许多基本问题仍未得到解答。我们展示了这些领域的许多成果。我们强调确定导致这种估计的数学条件，无论是黑洞还是普通物体。