Isoperimetric inequality

In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In formula_1-dimensional space formula_2 the inequality lower bounds the surface area or perimeter formula_3 of a set formula_4 by its volume formula_5,
where formula_7 is a unit sphere. The equality holds only when formula_8 is a sphere in formula_2.
On a plane, i.e. when formula_10, 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 formula_11, the isoperimetric inequality states, for the length "L" of a closed curve and the area "A" of the planar region that it encloses, that
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. 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.
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?
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". 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).
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
and that the equality holds if and only if the curve is a circle. The area of a disk of radius "R" is "πR"2 and the circumference of the circle is 2"πR", so both sides of the inequality are equal to 4"π"2"R"2 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. Equivalently, the isoperimetric ratio is at least 4 for every curve.
The isoperimetric quotient of a regular "n"-gon is
Let formula_16 be a smooth regular convex closed curve. Then the improved isoperimetric inequality states the following
where formula_18 denote the length of formula_16, the area of the region bounded by formula_16 and the oriented area of the Wigner caustic of formula_16, respectively, and the equality holds if and only if formula_16 is a curve of constant width.
On a sphere.
Let "C" be a simple closed curve on a sphere of radius 1. Denote by "L" the length of "C" and by "A" the area enclosed by "C". The spherical isoperimetric inequality states that
and that the equality holds if and only if the curve is a circle. There are, in fact, two ways to measure the spherical area enclosed by a simple closed curve, but the inequality is symmetric with the respect to taking the complement.
This inequality was discovered by Paul Lévy (1919) who also extended it to higher dimensions and general surfaces.
In the more general case of arbitrary radius "R", it is known that
In formula_2.
The isoperimetric inequality states that a sphere has the smallest surface area per given volume. Given a bounded set formula_26 with surface area formula_3 and volume formula_5, the isoperimetric inequality states
where formula_30 is a unit ball. The equality holds when formula_8 is a ball in formula_32. Under additional restrictions on the set (such as convexity, regularity, smooth boundary), the equality holds for a ball only. But in full generality the situation is more complicated. The relevant result of (for a simpler proof see ) is clarified in as follows. An extremal set consists of a ball and a "corona" that contributes neither to the volume nor to the surface area. That is, the equality holds for a compact set formula_8 if and only if formula_8 contains a closed ball formula_35 such that formula_36 and formula_37 For example, the "corona" may be a curve.
The proof of the inequality follows directly from Brunn–Minkowski inequality between a set formula_8 and a ball with radius formula_39, i.e. formula_40. By taking Brunn–Minkowski inequality to the power formula_1, subtracting formula_5 from both sides, dividing them by formula_39, and taking the limit as formula_44 (; ).
In full generality , the isoperimetric inequality states that for any set formula_4 whose closure has finite Lebesgue measure
where formula_47 is the ("n"-1)-dimensional Minkowski content, "Ln" is the "n"-dimensional Lebesgue measure, and "ωn" is the volume of the unit ball in formula_2. If the boundary of "S" is rectifiable, then the Minkowski content is the ("n"-1)-dimensional Hausdorff measure.
The "n"-dimensional isoperimetric inequality is equivalent (for sufficiently smooth domains) to the Sobolev inequality on formula_2 with optimal constant:
for all formula_51.
In Hadamard manifolds.
Hadamard manifolds are complete simply connected manifolds with nonpositive curvature. Thus they generalize the Euclidean space formula_2, which is a Hadamard manifold with curvature zero. In 1970's and early 80's, Thierry Aubin, Misha Gromov, Yuri Burago, and Viktor Zalgaller conjectured that the Euclidean isoperimetric inequality
holds for bounded sets formula_8 in Hadamard manifolds, which has become known as the Cartan–Hadamard conjecture.
In dimension 2 this had already been established in 1926 by André Weil, who was a student of Hadamard at the time.
In dimensions 3 and 4 the conjecture was proved by Bruce Kleiner in 1992, and Chris Croke in 1984 respectively.
In a metric measure space.
Most of the work on isoperimetric problem has been done in the context of smooth regions in Euclidean spaces, or more generally, in Riemannian manifolds. However, the isoperimetric problem can be formulated in much greater generality, using the notion of "Minkowski content". Let formula_55 be a "metric measure space": "X" is a metric space with metric "d", and "μ" is a Borel measure on "X". The "boundary measure", or Minkowski content, of a measurable subset "A" of "X" is defined as the lim inf
where
is the ε-"extension" of "A".
The isoperimetric problem in "X" asks how small can formula_58 be for a given "μ"("A"). If "X" is the Euclidean plane with the usual distance and the Lebesgue measure then this question generalizes the classical isoperimetric problem to planar regions whose boundary is not necessarily smooth, although the answer turns out to be the same.
The function
is called the "isoperimetric profile" of the metric measure space formula_55. Isoperimetric profiles have been studied for Cayley graphs of discrete groups and for special classes of Riemannian manifolds (where usually only regions "A" with regular boundary are considered).
For graphs.
In graph theory, isoperimetric inequalities are at the heart of the study of expander graphs, which are sparse graphs that have strong connectivity properties. Expander constructions have spawned research in pure and applied mathematics, with several applications to complexity theory, design of robust computer networks, and the theory of error-correcting codes.
Isoperimetric inequalities for graphs relate the size of vertex subsets to the size of their boundary, which is usually measured by the number of edges leaving the subset (edge expansion) or by the number of neighbouring vertices (vertex expansion). For a graph formula_61 and a number formula_62, the following are two standard isoperimetric parameters for graphs.
Here formula_65 denotes the set of edges leaving formula_8 and formula_67 denotes the set of vertices that have a neighbour in formula_8. The isoperimetric problem consists of understanding how the parameters formula_69 and formula_70 behave for natural families of graphs.
Example: Isoperimetric inequalities for hypercubes.
The formula_71-dimensional hypercube formula_72 is the graph whose vertices are all Boolean vectors of length formula_71, that is, the set formula_74. Two such vectors are connected by an edge in formula_72 if they are equal up to a single bit flip, that is, their Hamming distance is exactly one.
The following are the isoperimetric inequalities for the Boolean hypercube.
Edge isoperimetric inequality.
The edge isoperimetric inequality of the hypercube is formula_76. This bound is tight, as is witnessed by each set formula_8 that is the set of vertices of any subcube of formula_72.
Vertex isoperimetric inequality.
Harper's theorem says that "Hamming balls" have the smallest vertex boundary among all sets of a given size. Hamming balls are sets that contain all points of Hamming weight at most formula_79 and no points of Hamming weight larger than formula_80 for some integer formula_79. This theorem implies that any set formula_82 with
satisfies
As a special case, consider set sizes formula_85 of the form
for some integer formula_79. Then the above implies that the exact vertex isoperimetric parameter is
Isoperimetric inequality for triangles.
The isoperimetric inequality for triangles in terms of perimeter "p" and area "T" states that
with equality for the equilateral triangle. This is implied, via the AM–GM inequality, by a stronger inequality which has also been called the isoperimetric inequality for triangles: