Isoperimetry

Isoperimetry literally means "having an equal perimeter". In mathematics, isoperimetry is the general study of geometric figures having equal boundaries.

The isoperimetric problem in the plane

The classical isoperimetric problem dates back to antiquity. The problem can be stated as follows: Among all closed curves in the plane of fixed perimeter, which curve (if any) maximizes the area of its enclosed region? This question can be shown to be equivalent to the following problem: Among all closed curves in the plane enclosing a fixed area, which curve (if any) minimizes the perimeter?

Although the circle appears to be an obvious solution to the problem, proving this fact is rather difficult. The first progress toward the solution was made by Jakob Steiner in 1838, using a geometric method later named Steiner symmetrisation. Steiner showed that if a solution existed, then it must be the circle. Steiner's proof was completed later by several other mathematicians.

The theorem is usually stated in the form of an inequality that relates the perimeter and area of a closed curve in the plane. If P is the perimeter of the curve and A is the area of the region enclosed by the curve, then the inequality states that

<math>4\pi A \le P^2.<math>

For the case of a circle of radius r, we have A = πr² and P = 2πr, and substituting these into the inequality shows that the circle does indeed maximize the area among all curves of fixed perimeter. In fact, the circle is the only curve that maximizes the area.

There are dozens of proofs of this classic inequality. Several of these are discussed in the Treiberg paper below. In 1901, Hurwitz gave a purely analytic proof of the classical isoperimetric inequality based on Fourier series and Green's theorem.

The isoperimetric theorem generalises to higher dimensional spaces: the domain with volume 1 with the minimal surface area is always a ball.

See also