Isoperimetric Inequality - The Isoperimetric Inequality

The Isoperimetric Inequality

The solution to the isoperimetric problem is usually expressed in the form of an 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

and that the equality holds if and only if the curve is a circle. Indeed, the area of a disk of radius R is πR2 and the circumference of the circle is 2πR, so both sides of the inequality are equal to 4π2R2 in this case.

Dozens of proofs of the isoperimetric inequality have been found. In 1902, Hurwitz published a short proof using the Fourier series that applies to arbitrary rectifiable curves (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

and the isoperimetric inequality says that Q ≤ 1.

The isoperimetric quotient of a regular n-gon is