In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, geometric measure theory (GMT) is the study of
geometric properties of
sets (typically in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
) through
measure theory
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 simil ...
. It allows mathematicians to extend tools from
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
to a much larger class of
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
s that are not necessarily
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
.
History
Geometric measure theory was born out of the desire to solve
Plateau's problem
In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The problem ...
(named after
Joseph Plateau
Joseph Antoine Ferdinand Plateau (14 October 1801 – 15 September 1883) was a Belgian physicist and mathematician. He was one of the first people to demonstrate the illusion of a moving image. To do this, he used counterrotating disks with repea ...
) which asks if for every smooth closed curve in
there exists a
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
of least
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
A shape or figure is a graphics, graphical representation of an obje ...
among all surfaces whose
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
*Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film
*Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
equals the given curve. Such surfaces mimic
soap film
Soap films are thin layers of liquid (usually water-based) surrounded by air. For example, if two soap bubbles come into contact, they merge and a thin film is created in between. Thus, foams are composed of a network of films connected by Platea ...
s.
The problem had remained open since it was posed in 1760 by
Lagrange. It was solved independently in the 1930s by
Jesse Douglas
Jesse Douglas (3 July 1897 – 7 September 1965) was an American mathematician and Fields Medalist known for his general solution to Plateau's problem.
Life and career
He was born to a Jewish family in New York City, the son of Sarah (née ...
and
Tibor Radó
Tibor Radó (June 2, 1895 – December 29, 1965) was a Hungarian mathematician who moved to the United States after World War I.
Biography
Radó was born in Budapest and between 1913 and 1915 attended the Polytechnic Institute, studying civ ...
under certain
topological
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
restrictions. In 1960
Herbert Federer
Herbert Federer (July 23, 1920 – April 21, 2010) was an American mathematician. He is one of the creators of geometric measure theory, at the meeting point of differential geometry and mathematical analysis.Parks, H. (2012''Remembering Herbert ...
and
Wendell Fleming
Wendell Helms Fleming (born March 7, 1928) is an American mathematician, specializing in geometrical analysis and stochastic differential equations.
Fleming received in 1951 his PhD under Laurence Chisholm Young at the University of Wisconsin†...
used the theory of
currents
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
with which they were able to solve the orientable Plateau's problem
analytically without topological restrictions, thus sparking geometric measure theory. Later
Jean Taylor
Jean Ellen Taylor (born 1944) is an American mathematician who is a professor emerita at Rutgers University and visiting faculty at Courant Institute of Mathematical Sciences of New York University.
Biography
Taylor was born in Northern Califo ...
after
Fred Almgren proved
Plateau's laws
Plateau's laws describe the structure of soap films. These laws were formulated in the 19th century by the Belgian physicist Joseph Plateau from his experimental observations. Many patterns in nature are based on foams obeying these laws.
Laws ...
for the kind of singularities that can occur in these more general soap films and soap bubbles clusters.
Important notions
The following objects are central in geometric measure theory:
* Hausdorff measure and Hausdorff dimension
*
Rectifiable sets (or
Radon measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel se ...
s), which are
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
s with the least possible regularity required to admit approximate
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
s.
* Characterization of rectifiability through existence of approximate tangents, densities, projections, etc.
* Orthogonal projections, Besicovitch sets, Kakeya sets
* Uniform rectifiability
* Rectifiability and uniform rectifiability of (subsets of)
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 settin ...
s, e.g. SubRiemannian manifolds, Carnot groups, Heisenberg groups, etc.
* Connections to singular integrals, Fourier transform, Frostman measures, harmonic measures, etc
*
Currents
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
, a generalization of the concept of
oriented manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s, possibly with
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
*Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film
*Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
.
* Flat chains, an alternative generalization of the concept of
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s, possibly with
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
*Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film
*Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
.
*
Caccioppoli sets (also known as sets of locally finite perimeter), a generalization of the concept of
manifolds
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a Ne ...
on which the
divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
applies.
* Plateau type minimization problems from calculus of variations
The following theorems and concepts are also central:
* The
area formula, which generalizes the concept of
change of variables
Change or Changing may refer to:
Alteration
* Impermanence, a difference in a state of affairs at different points in time
* Menopause, also referred to as "the change", the permanent cessation of the menstrual period
* Metamorphosis, or change, ...
in integration.
* The
coarea formula, which generalizes and adapts
Fubini's theorem
In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if th ...
to geometric measure theory.
* The
isoperimetric inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
, which states that the smallest possible
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 to ...
for a given
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
A shape or figure is a graphics, graphical representation of an obje ...
is that of a round
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 const ...
.
*
Flat convergence In mathematics, flat convergence is a notion for convergence of submanifolds of Euclidean space. It was first introduced by Hassler Whitney in 1957, and then extended to integral currents by Herbert Federer, Federer and Fleming in 1960. It forms ...
, which generalizes the concept of manifold convergence.
Examples
The
Brunn–Minkowski inequality for the ''n''-dimensional volumes of
convex bodies
In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non-empty interior.
A convex body K is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x lies in ...
''K'' and ''L'',
:
can be proved on a single page and quickly yields the classical
isoperimetric inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
. The Brunn–Minkowski inequality also leads to
Anderson's theorem in statistics. The proof of the Brunn–Minkowski inequality predates modern measure theory; the development of measure theory and
Lebesgue integration
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Leb ...
allowed connections to be made between geometry and analysis, to the extent that in an integral form of the Brunn–Minkowski inequality known as the
Prékopa–Leindler inequality In mathematics, the Prékopa–Leindler inequality is an integral inequality closely related to the reverse Young's inequality, the Brunn–Minkowski inequality and a number of other important and classical inequalities in analysis. The result is ...
the geometry seems almost entirely absent.
See also
*
Caccioppoli set
*
Coarea formula
*
Currents
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
*
Herbert Federer
Herbert Federer (July 23, 1920 – April 21, 2010) was an American mathematician. He is one of the creators of geometric measure theory, at the meeting point of differential geometry and mathematical analysis.Parks, H. (2012''Remembering Herbert ...
*
Osgood curve
In mathematical analysis, an Osgood curve is a non-self-intersecting curve that has positive area. Despite its area, it is not possible for such a curve to cover a convex set, distinguishing them from space-filling curves. Osgood curves are named ...
References
*. The first paper of
Federer and Fleming illustrating their approach to the theory of perimeters based on the theory of
currents
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
.
*
*
*
*
*
*
*.
*{{eom, title=Geometric measure theory, first=T.C. , last=O'Neil
External links
Peter Mörters' GMT page
Measure theory