Geometric measure theory

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In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows to extend tools from differential geometry to a much larger class of surfaces that are not necessarily smooth.

History[edit]

Geometric measure theory was born out of the desire to solve the Plateau problem which asks if for every smooth closed curve in \mathbb{R}^3 there exists a surface of least area among all surfaces whose boundary equals the given curve. Such surfaces mimic soap films.

The problem had remained open since it was posed in 1760 by Lagrange. It was solved independently in the 1930s by Jesse Douglas and Tibor Radó under certain topological restrictions. In 1960 Herbert Federer and Wendell Fleming used the theory of currents with which they were able to solve Plateau's problem analytically without topological restrictions, thus sparking geometric measure theory. Later Jean Taylor after Fred Almgren proved Plateau's laws for the kind of singularities that can occur in these more general soap films and soap bubbles clusters.

Important Notions[edit]

The following concepts are central in geometric measure theory:

Examples[edit]

The Brunn–Minkowski inequality for the n-dimensional volumes of convex bodies K and L,

\mathrm{vol} \big( (1 - \lambda) K + \lambda L \big)^{1/n} \geq (1 - \lambda) \mathrm{vol} (K)^{1/n} + \lambda \, \mathrm{vol} (L)^{1/n},

can be proved on a single page and quickly yields the classical isoperimetric inequality. 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 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[edit]

References[edit]

External links[edit]

  • Peter Mörters' GMT page [1]
  • Toby O'Neil's GMT page with references [2]