Geometric measure theory
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry 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 (named after Joseph Plateau) which asks if for every smooth closed curve in 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
Important notions
The following objects are central in geometric measure theory:
- Hausdorff measure and Hausdorff dimension
- Rectifiable sets (or Radon measures), which are sets with the least possible regularity required to admit approximate tangent spaces.
- Characterization of rectifiability through existence of approximate tangents, densities, projections, etc.
- Orthogonal projections, Kakeya sets, Besicovitch sets
- Uniform rectifiability
- Rectifiability and uniform rectifiability of (subsets of) metric spaces, 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 manifolds, possibly with boundary.
- Flat chains, an alternative generalization of the concept of manifolds, possibly with boundary.
- manifolds on which the divergence theoremapplies.
- 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 in integration.
- The coarea formula, which generalizes and adapts Fubini's theorem to geometric measure theory.
- The isoperimetric inequality, which states that the smallest possible circumference for a given area is that of a round circle.
- Flat convergence, which generalizes the concept of manifold convergence.
Examples
The
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
References
- Zbl 0187.31301. The first paper of Federer and Fleming illustrating their approach to the theory of perimeters based on the theory of currents.
- MR 0257325
- Federer, H. (1978), "Colloquium lectures on geometric measure theory", Bull. Amer. Math. Soc., 84 (3): 291–338,
- ISBN 978-0792302308
- Gardner, Richard J. (2002), "The Brunn-Minkowski inequality", Bull. Amer. Math. Soc. (N.S.), 39 (3): 355–405 (electronic), MR 1898210
- ISBN 978-0-521-65595-8
- MR 2455580
- MR 0428181.
- O'Neil, T.C. (2001) [1994], "Geometric measure theory", Encyclopedia of Mathematics, EMS Press