Editing Isoperimetric inequality (section)

== On a sphere ==

Let ''C'' be a simple closed curve on a [[sphere]] of radius 1. Denote by ''L'' the length of ''C'' and by ''A'' the area enclosed by ''C''. The '''spherical isoperimetric inequality''' states that

:<math>L^2 \ge A (4\pi - A),</math>

and that the equality holds if and only if the curve is a circle. There are, in fact, two ways to measure the spherical area enclosed by a simple closed curve, but the inequality is symmetric with the respect to taking the complement.

This inequality was discovered by [[Paul Lévy (mathematician)|Paul Lévy]] (1919) who also extended it to higher dimensions and general surfaces.<ref>{{Cite book|chapter-url=https://cds.cern.ch/record/1412861|title=Metric Structures for Riemannian and Non-Riemannian Spaces|last1=Gromov|first1=Mikhail|last2=Pansu|first2=Pierre|date=2006|publisher=Springer|isbn=9780817645830|series=Modern Birkhäuser Classics|location=Dordrecht|pages=519|chapter=Appendix C. Paul Levy's Isoperimetric Inequality}}</ref>

In the more general case of arbitrary radius ''R'', it is known<ref>[[Robert Osserman|Osserman, Robert]]. "The Isoperimetric Inequality." Bulletin of the American Mathematical Society. 84.6 (1978) http://www.ams.org/journals/bull/1978-84-06/S0002-9904-1978-14553-4/S0002-9904-1978-14553-4.pdf</ref> that

:<math>L^2\ge 4\pi A - \frac{A^2}{R^2}.</math>