Editing Isoperimetric inequality (section)

== On a plane ==
The solution to the isoperimetric problem is usually expressed in the form of an [[inequality (mathematics)|inequality]] that relates the length ''L'' of a closed curve and the area ''A'' of the planar region that it encloses. The '''isoperimetric inequality''' states that

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

and that the equality holds if and only if the curve is a circle. The [[area of a disk]] of radius ''R'' is ''πR''<sup>2</sup> and the circumference of the circle is 2''πR'', so both sides of the inequality are equal to 4''π''<sup>2</sup>''R''<sup>2</sup> in this case.

Dozens of proofs of the isoperimetric inequality have been found. In 1902, [[Adolf Hurwitz|Hurwitz]] published a short proof using the [[Fourier series]] that applies to arbitrary [[rectifiable curve]]s (not assumed to be smooth). An elegant direct proof based on comparison of a smooth simple closed curve with an appropriate circle was given by E. Schmidt in 1938. It uses only the [[arc length]] formula, expression for the area of a plane region from [[Green's theorem]], and the [[Cauchy–Schwarz inequality]].

For a given closed curve, the '''isoperimetric quotient''' is defined as the ratio of its area and that of the circle having the same perimeter. This is equal to

:<math>Q=\frac{4\pi A}{L^2}</math>

and the isoperimetric inequality says that ''Q'' ≤ 1. Equivalently, the [[isoperimetric ratio]] {{math|''L''<sup>2</sup>/''A''}} is at least 4{{pi}} for every curve.

The isoperimetric quotient of a regular ''n''-gon is

:<math>Q_n=\frac{\pi}{n \tan(\pi/n)}.</math>

Let <math>C</math> be a smooth regular convex closed curve. Then the '''improved isoperimetric inequality''' states the following

:<math>L^2\geqslant 4\pi A+8\pi\left|\widetilde{A}_{0.5}\right|,</math>

where <math>L, A, \widetilde{A}_{0.5}</math> denote the length of <math>C</math>, the area of the region bounded by <math>C</math> and the oriented area of the [[Wigner caustic]] of <math>C</math>, respectively, and the equality holds if and only if <math>C</math> is a [[curve of constant width]].<ref>{{cite journal| title = The improved isoperimetric inequality and the Wigner caustic of planar ovals | first = Michał | last = Zwierzyński | journal = J. Math. Anal. Appl. | volume = 442 | date = 2016| pages = 726–739|doi=10.1016/j.jmaa.2016.05.016|issue=2 | arxiv=1512.06684| s2cid = 119708226 }}</ref>