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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 Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
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'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
) 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 simila ...
. 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 mult ...
to a much larger class of surfaces that are not necessarily smooth.


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 repe ...
) which asks if for every smooth closed curve in \mathbb^3 there exists a surface 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 or planar lamina, while '' surface area'' refers to the area of an op ...
among all surfaces whose boundary 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 Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaJesse Douglas 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 ...
under certain topological 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 F ...
and Wendell Fleming used the theory of currents with which they were able to solve the orientable Plateau's problem analytically without topological restrictions, thus sparking geometric measure theory. Later Jean Taylor 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 measures), 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 setti ...
s, e.g. SubRiemannian manifolds, Carnot groups, Heisenberg groups, etc. * Connections to singular integrals, Fourier transform, Frostman measures, harmonic measures, etc * Currents, a generalization of the concept of
oriented In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
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. * 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. *
Caccioppoli set In mathematics, a Caccioppoli set is a set whose boundary is measurable and has (at least locally) a ''finite measure''. A synonym is set of (locally) finite perimeter. Basically, a set is a Caccioppoli set if its characteristic function is a f ...
s (also known as sets of locally finite perimeter), a generalization of the concept of manifolds 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 ...
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 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 ...
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 or planar lamina, while '' surface area'' refers to the area of an op ...
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 cons ...
. *
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 Federer and Fleming in 1960. It forms a fundamental ...
, which generalizes the concept of manifold convergence.


Examples

The Brunn–Minkowski inequality for the ''n''-dimensional volumes of convex bodies ''K'' and ''L'', :\mathrm \big( (1 - \lambda) K + \lambda L \big)^ \geq (1 - \lambda) \mathrm (K)^ + \lambda \, \mathrm (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 Le ...
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 the geometry seems almost entirely absent.


See also

*
Caccioppoli set In mathematics, a Caccioppoli set is a set whose boundary is measurable and has (at least locally) a ''finite measure''. A synonym is set of (locally) finite perimeter. Basically, a set is a Caccioppoli set if its characteristic function is a f ...
* Coarea formula * Currents *
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 F ...
* Osgood curve


References

*. The first paper of Federer and Fleming illustrating their approach to the theory of perimeters based on the theory of currents. * * * * * * *. *{{eom, title=Geometric measure theory, first=T.C. , last=O'Neil


External links


Peter Mörters' GMT page


Measure theory