Editing Isoperimetric inequality (section)

== The isoperimetric problem in the plane ==
[[File:Isoperimetric inequality illustr1.svg|right|thumb|If a region is not convex, a "dent" in its boundary can be "flipped" to increase the area of the region while keeping the perimeter unchanged.]]
[[File:Isoperimetric inequality illustr2.svg|right|thumb|An elongated shape can be made more round while keeping its perimeter fixed and increasing its area.]]

The classical ''isoperimetric problem'' dates back to antiquity.<ref>{{Cite web|last=Olmo|first=Carlos Beltrán, Irene|date=2021-01-04|title=Sobre mates y mitos|url=https://elpais.com/ciencia/2021-01-04/sobre-mates-y-mitos.html|access-date=2021-01-14|website=El País|language=es}}</ref> The problem can be stated as follows: Among all closed [[curve]]s in the plane of fixed perimeter, which curve (if any) maximizes the area of its enclosed region? This question can be shown to be equivalent to the following problem: Among all closed curves in the plane enclosing a fixed area, which curve (if any) minimizes the perimeter?

This problem is conceptually related to the [[principle of least action]] in [[physics]], in that it can be restated: what is the principle of action which encloses the greatest area, with the greatest economy of effort?{{citation needed|date=July 2024}} The 15th-century philosopher and scientist, Cardinal [[Nicholas of Cusa]], considered [[rotation]]al action, the process by which a [[circle]] is generated, to be the most direct reflection, in the realm of sensory impressions, of the process by which the universe is created. German astronomer and astrologer [[Johannes Kepler]] invoked the isoperimetric principle in discussing the morphology of the [[Solar System]], in ''[[Mysterium Cosmographicum]]'' (''The Sacred Mystery of the Cosmos'', 1596).

Although the circle appears to be an obvious solution to the problem, proving this fact is rather difficult. The first progress toward the solution was made by Swiss geometer [[Jakob Steiner]] in 1838, using a geometric method later named [[Symmetrization methods#Steiner Symmetrization|''Steiner symmetrisation'']].<ref>J. Steiner, ''Einfacher Beweis der isoperimetrischen Hauptsätze'', J. reine angew Math. '''18''', (1838), pp. 281&ndash;296; and Gesammelte Werke Vol. 2, pp. 77&ndash;91, Reimer, Berlin, (1882).</ref> Steiner showed that if a solution existed, then it must be the circle. Steiner's proof was completed later by several other mathematicians.

Steiner begins with some geometric constructions which are easily understood; for example, it can be shown that any closed curve enclosing a region that is not fully [[Convex set|convex]] can be modified to enclose more area, by "flipping" the concave areas so that they become convex. It can further be shown that any closed curve which is not fully symmetrical can be "tilted" so that it encloses more area. The one shape that is perfectly convex and symmetrical is the circle, although this, in itself, does not represent a rigorous proof of the isoperimetric theorem (see external links).