Isoperimetric inequality

In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In ${\displaystyle n}$-dimensional space ${\displaystyle \mathbb {R} ^{n}}$ the inequality lower bounds the surface area or perimeter ${\displaystyle \operatorname {per} (S)}$ of a set ${\displaystyle S\subset \mathbb {R} ^{n}}$ by its volume ${\displaystyle \operatorname {vol} (S)}$,

${\displaystyle \operatorname {per} (S)\geq n\operatorname {vol} (S)^{(n-1)/n}\,\operatorname {vol} (B_{1})^{1/n}}$,

where ${\displaystyle B_{1}\subset \mathbb {R} ^{n}}$ is a unit sphere. The equality holds only when ${\displaystyle S}$ is a sphere in ${\displaystyle \mathbb {R} ^{n}}$.

On a plane, i.e. when ${\displaystyle n=2}$, 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 ${\displaystyle \mathbb {R} ^{2}}$, the isoperimetric inequality states, for the length L of a closed curve and the area A of the planar region that it encloses, that

${\displaystyle L^{2}\geq 4\pi A,}$

and that equality holds if and only if the curve is a circle.

The isoperimetric problem is to determine a plane figure of the largest possible area whose boundary has a specified length.^[1] The closely related Dido's problem asks for a region of the maximal area bounded by a straight line and a curvilinear arc whose endpoints belong to that line. It is named after Dido, the legendary founder and first queen of Carthage. The solution to the isoperimetric problem is given by a circle and was known already in Ancient Greece. However, the first mathematically rigorous proof of this fact was obtained only in the 19th century. Since then, many other proofs have been found.

The isoperimetric problem has been extended in multiple ways, for example, to curves on surfaces and to regions in higher-dimensional spaces. Perhaps the most familiar physical manifestation of the 3-dimensional isoperimetric inequality is the shape of a drop of water. Namely, a drop will typically assume a symmetric round shape. Since the amount of water in a drop is fixed, surface tension forces the drop into a shape which minimizes the surface area of the drop, namely a round sphere.