Isoperimetric inequality

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

{\displaystyle \operatorname {per} (S)\geq n\operatorname {vol} (S)^{\frac {n-1}{n}},\operatorname {vol} (B_{1})^{\frac {1}{n}}}{\displaystyle \operatorname {per} (S)\geq n\operatorname {vol} (S)^{\frac {n-1}{n}},\operatorname {vol} (B_{1})^{\frac {1}{n}}},
where {\displaystyle B_{1}\subset \mathbb {R} ^{n}}{\displaystyle B_{1}\subset \mathbb {R} ^{n}} is a unit sphere. The equality holds only when {\displaystyle S}S is a sphere in {\displaystyle \mathbb {R} ^{n}}\mathbb {R} ^{n}.

On a plane, i.e. when {\displaystyle n=2}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}}{\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,}{\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.

1 The isoperimetric problem in the plane

The classical isoperimetric problem dates back to antiquity.[2] 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.[3] 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).

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.

2 On a plane

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

{\displaystyle 4\pi A\leq L^{2},}4\pi A\leq L^{2},
and that the equality holds if and only if the curve is a circle. 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

{\displaystyle Q={\frac {4\pi A}{L^{2}}}}Q={\frac {4\pi A}{L^{2}}}
and the isoperimetric inequality says that Q ≤ 1. Equivalently, the isoperimetric ratio L2/A is at least 4π for every curve.

The isoperimetric quotient of a regular n-gon is

{\displaystyle Q_{n}={\frac {\pi }{n\tan {\tfrac {\pi }{n}}}}.}Q_{n}={\frac {\pi }{n\tan {\tfrac {\pi }{n}}}}.
Let {\displaystyle C}C be a smooth regular convex closed curve. Then the improved isoperimetric inequality states the following

{\displaystyle L^{2}\geqslant 4\pi A+8\pi \left|{\widetilde {A}}{0.5}\right|,}{\displaystyle L^{2}\geqslant 4\pi A+8\pi \left|{\widetilde {A}}{0.5}\right|,}
where {\displaystyle L,A,{\widetilde {A}}{0.5}}{\displaystyle L,A,{\widetilde {A}}{0.5}} denote the length of {\displaystyle C}C, the area of the region bounded by {\displaystyle C}C and the oriented area of the Wigner caustic of {\displaystyle C}C, respectively, and the equality holds if and only if {\displaystyle C}C is a curve of constant width.[4]

3 On a sphere

4 In Rn

5 In Hadamard manifolds

6 In a metric measure space

7 For graphs

7.1 Example: Isoperimetric inequalities for hypercubes

7.1.1 Edge isoperimetric inequality

7.1.2 Vertex isoperimetric inequality

8 Isoperimetric inequality for triangles

9 See also