1 Answers
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n {\displaystyle n} -dimensional space R n {\displaystyle \mathbb {R} ^{n}} the inequality lower bounds the surface area or perimeter per {\displaystyle \operatorname {per} } of a set S ⊂ R n {\displaystyle S\subset \mathbb {R} ^{n}} by its volume vol {\displaystyle \operatorname {vol} } ,
where B 1 ⊂ R n {\displaystyle B_{1}\subset \mathbb {R} ^{n}} is a unit sphere. The equality holds only when S {\displaystyle S} is a sphere in R n {\displaystyle \mathbb {R} ^{n}}.
On a plane, i.e. when n = 2 {\displaystyle n=2} , the isoperimetric inequality relates the square of the circumference of a closed curve and the area of a plane region it encloses. Isoperimetric literally means "having the same perimeter". Specifically in R 2 {\displaystyle \mathbb {R} ^{2}} , the isoperimetric inequality states, for the length L of a closed curve and the area A of the planar region that it encloses, that
and that equality holds if and only if the curve is a circle.