Editing Isoperimetric inequality (section)

{{short description|Geometric inequality applicable to any closed curve}}
{{Use dmy dates|date=May 2023}}
In [[mathematics]], the '''isoperimetric inequality''' is a [[geometry|geometric]] [[inequality (mathematics)|inequality]] involving the square of the [[circumference]] of a [[closed curve]] in the plane and the [[area]] of a plane region it encloses, as well as its various generalizations. ''[[wikt:isoperimetric#English|Isoperimetric]]'' literally means "having the same [[perimeter]]". Specifically, the isoperimetric inequality states, for the length ''L'' of a closed curve and the area ''A'' of the planar region that it encloses, that

:<math>4\pi A \le L^2,</math>

and that equality holds [[if and only if]] the curve is a circle.

The '''isoperimetric problem''' is to determine a [[plane figure]] of the largest possible area whose [[boundary (topology)|boundary]] has a specified length.<ref>{{cite journal|author=Blåsjö, Viktor|title=The Evolution of the Isoperimetric Problem|journal=Amer. Math. Monthly|volume=112|year=2005|pages=526–566|doi=10.1080/00029890.2005.11920227 |url=http://www.maa.org/programs/maa-awards/writing-awards/the-evolution-of-the-isoperimetric-problem}}</ref> The closely related ''Dido's problem'' asks for a region of the maximal area bounded by a straight line and a curvilinear [[arc (geometry)|arc]] whose endpoints belong to that line. It is named after [[Dido (Queen of Carthage)|Dido]], the legendary founder and first queen of [[Carthage]]. The solution to the isoperimetric problem is given by a [[circle]] and was known already in [[Ancient Greece]]. However, the first mathematically rigorous proof of this fact was obtained only in the 19th century. Since then, many other proofs have been found.

The isoperimetric problem has been extended in multiple ways, for example, to curves on [[differential geometry of surfaces|surfaces]] and to regions in higher-dimensional spaces. Perhaps the most familiar physical manifestation of the 3-dimensional isoperimetric inequality is the shape of a drop of water. Namely, a drop will typically assume a symmetric round shape. Since the amount of water in a drop is fixed, [[surface tension]] forces the drop into a shape which minimizes the surface area of the drop, namely a round sphere.