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 Rn{\displaystyle \mathbb {R} ^{n}} the inequality lower bounds the surface area surf(S){\displaystyle \mathrm {surf} (S)} of a set S⊂Rn{\displaystyle S\subset \mathbb {R} ^{n}} by its volume vol(S){\displaystyle \mathrm {vol} (S)},