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

If a region is not convex, a "dent" in its boundary can be "flipped" to increase the area of the region while keeping the perimeter unchanged.

An elongated shape can be made more round while keeping its perimeter fixed and increasing its area.

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?

This problem is conceptually related to the principle of least action in physics, in that it can be restated: what is the principle of action which encloses the greatest area, with the greatest economy of effort? The 15th-century philosopher and scientist, Cardinal Nicholas of Cusa, considered rotational action, the process by which a circle is generated, to be the most direct reflection, in the realm of sensory impressions, of the process by which the universe is created. German astronomer and astrologer Johannes Kepler invoked the isoperimetric principle in discussing the morphology of the solar system, in Mysterium Cosmographicum (The Sacred Mystery of the Cosmos, 1596).

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 Swiss geometer Jakob Steiner in 1838, using a geometric method later named Steiner symmetrisation.^{[citation needed]} Steiner showed that if a solution existed, then it must be the circle. Steiner's proof was completed later by several other mathematicians.

Steiner begins with some geometric constructions which are easily understood; for example, it can be shown that any closed curve enclosing a region that is not fully convex can be modified to enclose more area, by "flipping" the concave areas so that they become convex. It can further be shown that any closed curve which is not fully symmetrical can be "tilted" so that it encloses more area. The one shape that is perfectly convex and symmetrical is the circle, although this, in itself, does not represent a rigorous proof of the isoperimetric theorem (see external links).

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

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.

A stronger version of the inequality is Bonnesen's inequality, giving an estimate on the isoperimetric defect.

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.

Modern formulations of isoperimetric problems are sometimes given in terms of sub-Riemannian geometry; Dido's problem specifically finds expression in terms of the Heisenberg group: given an arc connecting two points, the "height" z of a point in the Heisenberg group corresponds to the area subtended by the arc.

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

Let be a metric measure space (d is a metric on X and μ is a Borel measure). For , let

where

is the ε-extension of A.

(The number is called the boundary measure of A in the sense of Minkowski, associated with μ.)

Then the (Minkowski) isoperimetric problem in A asks how small can be, if μ(A) is given. More formally, the function is called the isoperimetric profile of .

For example, if , is the Lebesgue measure and d the Euclidean metric, the problem coincides with the isoperimetric problem in the plane, discussed above.

• Isoperimetric dimension
• Dido's problem
• Gaussian isoperimetric inequality
• Spherical isoperimetric inequality
• Lévy-Gromov inequality
• Expander graph
• Sobolev inequality

• History of the Isoperimetric Problem at Convergence
• Treiberg: Several proofs of the isoperimetric inequality

• Isoperimetric Theorem at cut-the-knot