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

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The inequality (\[eq:54\]) applies to complicated geometric objects (i.e. arbitrary closed planar curves). The equality in (\[eq:54\]) is achieved only for an object of “optimal shape” (i.e. the circle) which is described by few parameters (in this case only one: the radius). Moreover, this object has a variational characterization: the circle is uniquely characterized by the property that among all simple closed plane curves of given length $L$, the circle of circumference $L$ encloses the maximum area.

General Relativity is a geometric theory, hence it is not surprising that geometric inequalities appear naturally in it. Many of these inequalities are similar in spirit as the isoperimetric inequality (\[eq:54\]). In particular, all the geometric inequalities discussed in this article will have the same structure as (\[eq:54\]): the inequality applies for a rich class of objects and the equality only applies for an object of “optimal shape” (always indicated in parenthesis as in (\[eq:54\])). This object, like the circle, can be described by few parameters and it has also a variational characterization.

However, General Relativity is also a physical 