hole (i.e. the Kerr metric) also describe naked singularities. In effect, the Kerr metric depends on two parameters: the mass $m$ and the angular momentum $J$. This metric is a solution of Einstein vacuum equations for any choice of the parameters $m$ and $J$. However, it represents a black hole if and only if the following remarkably inequality holds $$\label{eq:2}
\sqrt{|J|}\leq  m.$$ Otherwise the spacetime contains a naked singularity. Figure \[fig:2\] shows the parameter space of the Kerr solution. Extreme black holes are defined by the equality in (\[eq:2\]). These black holes lie at the boundary between naked singularities and black holes. For most of the inequalities discussed in this article, extreme black holes play the role of the circle in the isoperimetric inequality (\[eq:54\]): they reach the equality and they represent objects of “optimal shape”.

![A point in this graph is a Kerr solution with parameters $m$ and $J$. The horizontal axis where $m=0$ is Minkowski space. The Schwarzschild solution is given by the vertical axis where $J=0$. In the gray region the parameters satisfy the inequality (\[eq:2\]) and hence the Kerr solution describe a black hole. The boundary of this region is given by the equality in (\[eq:2\]), 