---
abstract: 'It is well known that the three parameters that characterize the Kerr black hole (mass, angular momentum and horizon area) satisfy several important inequalities. Remarkably, some of these inequalities remain valid also for dynamical black holes. This kind of inequalities play an important role in the characterization of the gravitational collapse. They are closed related with the cosmic censorship conjecture. In this article recent results in this subject are reviewed.'
author:
- |
    Sergio Dain\
    Facultad de Matemática, Astronomía y Física, FaMAF,\
    Universidad Nacional de Córdoba,\
    Instituto de Física Enrique Gaviola, IFEG, CONICET,\
    Ciudad Universitaria (5000) Córdoba, Argentina.
title: Geometric inequalities for black holes
---

Geometric inequalities in General Relativity {#sec:geom-ineq}
============================================

A classical example of a geometric inequality is the isoperimetric inequality for closed plane curves given by $$\label{eq:54}
  L^2 \geq 4\pi A\quad (=\text{ circle}),$$ where $A$ is the area enclosed by a curve $C$ of length $L$. In (\[eq:54\]) equality holds if and only if $C$ is a circle, see figure \[fig:1\]. For a review on this subject see [@Osserman78].

![The isoperimetric inequality. On the left an arbitrary curve, where the strict inequality holds. On the right the circle, where the equality holds.](isoperimetric-curve-p.pdf "fig:"){width="3cm"} ![The isoperimetric 