In mathematics, the isoperimetric inequality is a
geometric inequality
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
involving the perimeter of a set and its volume. In
-dimensional space
the inequality lower bounds the
surface area or
perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pr ...
of a set
by its
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
,
:
,
where
is a
unit sphere
In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit ...
. The equality holds only when
is a sphere in
.
On a plane, i.e. when
, the isoperimetric inequality relates the square of the
circumference
In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out t ...
of a
closed curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
and the
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ope ...
of a plane region it encloses. ''
Isoperimetric'' literally means "having the same
perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pr ...
". Specifically in
, 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
A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type.
A plane shape or plane figure is constrained to lie o ...
of the largest possible area whose
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
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
Dido ( ; , ), also known as Elissa ( , ), was the legendary founder and first queen of the Phoenician city-state of Carthage (located in modern Tunisia), in 814 BC.
In most accounts, she was the queen of the Phoenician city-state of Tyre (t ...
, the legendary founder and first queen of
Carthage
Carthage was the capital city of Ancient Carthage, on the eastern side of the Lake of Tunis in what is now Tunisia. Carthage was one of the most important trading hubs of the Ancient Mediterranean and one of the most affluent cities of the cla ...
. The solution to the isoperimetric problem is given by a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
and was known already in
Ancient Greece
Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean Sea, Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, classical antiquity ( AD 600), th ...
. 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
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space.
Surface or surfaces may also refer to:
Mathematics
*Surface (mathematics), a generalization of a plane which needs not be flat
* Sur ...
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
![Isoperimetric inequality illustr2](https://upload.wikimedia.org/wikipedia/commons/d/d4/Isoperimetric_inequality_illustr2.svg)
The classical ''isoperimetric problem'' dates back to antiquity. The problem can be stated as follows: Among all closed
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s 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
The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the '' action'' of a mechanical system, yields the equations of motion for that system. The principle states tha ...
in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, 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
Nicholas of Cusa (1401 – 11 August 1464), also referred to as Nicholas of Kues and Nicolaus Cusanus (), was a German Catholic cardinal, philosopher, theologian, jurist, mathematician, and astronomer. One of the first German proponents of Re ...
, considered
rotational action, the process by which a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
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
''Mysterium Cosmographicum'' (lit. ''The Cosmographic Mystery'', alternately translated as ''Cosmic Mystery'', ''The Secret of the World'', or some variation) is an astronomy book by the German astronomer Johannes Kepler, published at Tübingen i ...
'' (''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
Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry.
Life
Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards st ...
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
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
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
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
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
In geometry, the area enclosed by a circle of radius is . Here the Greek letter represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.
One method of deriving this formula, which origi ...
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
Hurwitz is one of the variants of a surname of Ashkenazi Jewish origin (for historical background see the Horowitz page).
Notable people with the surname include:
*Adolf Hurwitz (1859–1919), German mathematician
** Hurwitz polynomial
**Hurwitz m ...
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
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
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 In analytic geometry, the isoperimetric ratio of a simple closed curve in the Euclidean plane is the ratio , where is the length of the curve and is its area. It is a dimensionless quantity that is invariant under similarity transformations of t ...
is at least 4 for every curve.
The isoperimetric quotient of a regular ''n''-gon is
:
Let
be a smooth regular convex closed curve. Then the improved isoperimetric inequality states the following
:
where
denote the length of
, the area of the region bounded by
and the oriented area of the
Wigner caustic of
, respectively, and the equality holds if and only if
is a
curve of constant width
In geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of constant width ...
.
On a sphere
Let ''C'' be a simple closed curve on a
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
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
The isoperimetric inequality states that a
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
has the smallest surface area per given volume. Given a bounded set
with
surface area and
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
, the isoperimetric inequality states
:
where
is a
unit ball
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
. The equality holds when
is a ball in
. Under additional restrictions on the set (such as
convexity
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
,
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
if and only if
contains a closed ball
such that
and
For example, the "corona" may be a curve.
The proof of the inequality follows directly from
Brunn–Minkowski inequality between a set
and a ball with radius
, i.e.
. By taking Brunn–Minkowski inequality to the power
, subtracting
from both sides, dividing them by
, and taking the limit as
(; ).
In full generality , the isoperimetric inequality states that for any set
whose
closure has finite
Lebesgue measure
:
where
is the (''n''-1)-dimensional
Minkowski content, ''L
n'' is the ''n''-dimensional Lebesgue measure, and ''ω
n'' is the volume of the
unit ball
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
in
. If the boundary of ''S'' is
rectifiable, then the Minkowski content is the (''n''-1)-dimensional
Hausdorff measure
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that as ...
.
The ''n''-dimensional isoperimetric inequality is equivalent (for sufficiently smooth domains) to the
Sobolev inequality
In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the ...
on
with optimal constant:
:
for all
.
In Hadamard manifolds
Hadamard manifolds are complete simply connected manifolds with nonpositive curvature. Thus they generalize the Euclidean space
, which is a Hadamard manifold with curvature zero. In 1970's and early 80's,
Thierry Aubin
Thierry Aubin (6 May 1942 – 21 March 2009) was a French mathematician who worked at the Centre de Mathématiques de Jussieu, and was a leading expert on Riemannian geometry
and non-linear partial differential equations. His fundamental contrib ...
,
Misha Gromov,
Yuri Burago
Yuri Dmitrievich Burago (russian: Ю́рий Дми́триевич Бура́го) (born 1936) is a Russian mathematician. He works in differential and convex geometry.
Education and career
Burago studied at Leningrad University, where he ...
, and
Viktor Zalgaller conjectured that the Euclidean isoperimetric inequality
:
holds for bounded sets
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
Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations.
Biography
The son of a teac ...
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 space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
s, or more generally, in
Riemannian manifolds. However, the isoperimetric problem can be formulated in much greater generality, using the notion of ''Minkowski content''. Let
be a ''metric measure space'': ''X'' is a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
with
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
''d'', and ''μ'' is a
Borel measure on ''X''. The ''boundary measure'', or
Minkowski content, of a
measurable
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
subset ''A'' of ''X'' is defined as the
lim inf
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a ...
:
where
:
is the ε-''extension'' of ''A''.
The isoperimetric problem in ''X'' asks how small can
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
. Isoperimetric profiles have been studied for
Cayley graph
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cay ...
s of
discrete group
In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and o ...
s and for special classes of Riemannian manifolds (where usually only regions ''A'' with regular boundary are considered).
For graphs
In
graph theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, isoperimetric inequalities are at the heart of the study of
expander graphs
In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander constructions have spawned research in pure and applied mathematics, with several applica ...
, which are
sparse graph
In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges (where every pair of vertices is connected by one edge). The opposite, a graph with only a few edges, is a sparse graph. The distinction ...
s 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 network
A computer network is a set of computers sharing resources located on or provided by network nodes. The computers use common communication protocols over digital interconnections to communicate with each other. These interconnections are ...
s, 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
and a number
, the following are two standard isoperimetric parameters for graphs.
*The edge isoperimetric parameter:
::
*The vertex isoperimetric parameter:
::
Here
denotes the set of edges leaving
and
denotes the set of vertices that have a neighbour in
. The isoperimetric problem consists of understanding how the parameters
and
behave for natural families of graphs.
Example: Isoperimetric inequalities for hypercubes
The
-dimensional
hypercube is the graph whose vertices are all Boolean vectors of length
, that is, the set
. Two such vectors are connected by an edge in
if they are equal up to a single bit flip, that is, their
Hamming distance
In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of ''substitutions'' required to chan ...
is exactly one.
The following are the isoperimetric inequalities for the Boolean hypercube.
Edge isoperimetric inequality
The edge isoperimetric inequality of the hypercube is
. This bound is tight, as is witnessed by each set
that is the set of vertices of any subcube of
.
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
The Hamming weight of a string is the number of symbols that are different from the zero-symbol of the alphabet used. It is thus equivalent to the Hamming distance from the all-zero string of the same length. For the most typical case, a string ...
at most
and no points of Hamming weight larger than
for some integer
. This theorem implies that any set
with
:
satisfies
:
As a special case, consider set sizes
of the form
:
for some integer
. 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
[Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in ''Mathematical Plums'' (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.]
:
with equality for the
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
. This is implied, via the
AM–GM inequality
In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; a ...
, by a stronger inequality which has also been called the isoperimetric inequality for triangles:
[Dragutin Svrtan and Darko Veljan, "Non-Euclidean Versions of Some Classical Triangle Inequalities", ''Forum Geometricorum'' 12, 2012, 197–209. http://forumgeom.fau.edu/FG2012volume12/FG201217.pdf]
:
See also
*
Blaschke–Lebesgue theorem
In plane geometry the Blaschke–Lebesgue theorem states that the Reuleaux triangle has the least area of all curves of given constant width. In the form that every curve of a given width has area at least as large as the Reuleaux triangle, it is ...
*
Chaplygin problem
*
Curve-shortening flow
In mathematics, the curve-shortening flow is a process that modifies a smooth curve in the Euclidean plane by moving its points perpendicularly to the curve at a speed proportional to the curvature. The curve-shortening flow is an example of a g ...
*
Expander graph
In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander constructions have spawned research in pure and applied mathematics, with several applica ...
*
Gaussian isoperimetric inequality
*
Isoperimetric dimension
*
Isoperimetric point
*
List of triangle inequalities
In geometry, triangle inequalities are inequalities involving the parameters of triangles, that hold for every triangle, or for every triangle meeting certain conditions. The inequalities give an ordering of two different values: they are of th ...
*
Planar separator theorem
In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split into smaller pieces by removing a small number of vertices. Specifically, the removal of verti ...
*
Mixed volume
Notes
References
*
Blaschke and Leichtweiß, ''Elementare Differentialgeometrie'' (in German), 5th edition, completely revised by K. Leichtweiß. Die Grundlehren der mathematischen Wissenschaften, Band 1.
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, New York Heidelberg Berlin, 1973
*
*
*
*
*
*
*.
*
Gromov, M.: "Paul Levy's isoperimetric inequality". Appendix C in ''Metric structures for Riemannian and non-Riemannian spaces''. Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. Birkhäuser Boston, Inc., Boston, Massachusetts, 1999.
*.
*
*
*.
*
*.
*.
External links
{{commons category, Isoperimetric inequality
History of the Isoperimetric Problema
ConvergenceTreiberg: Several proofs of the isoperimetric inequalityIsoperimetric Theoremat
cut-the-knot
Multivariable calculus
Calculus of variations
Geometric inequalities
Analytic geometry