Isoperimetric inequality

In mathematics, the isoperimetric inequality is a geometric inequality involving the surface area 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 s u r f ( S ) {displaystyle mathrm {surf} (S)} of a set S ⊂ R n {displaystyle Ssubset mathbb {R} ^{n}} by its volume v o l ( S ) {displaystyle mathrm {vol} (S)} , In mathematics, the isoperimetric inequality is a geometric inequality involving the surface area 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 s u r f ( S ) {displaystyle mathrm {surf} (S)} of a set S ⊂ R n {displaystyle Ssubset mathbb {R} ^{n}} by its volume v o l ( S ) {displaystyle mathrm {vol} (S)} , where B 1 ⊂ R n {displaystyle B_{1}subset mathbb {R} ^{n}} is a unit ball. The equality holds when S {displaystyle S} is a ball in R n {displaystyle mathbb {R} ^{n}} . On a plane, i.e. when n = 2 {displaystyle n=2} , the isoperimetric inequality relates 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