Geometric measure theory

Geometric measure theory

In mathematics, geometric measure theory (GMT) is the study of the geometric properties of the measures of sets (typically in Euclidean spaces), including such things as arc lengths and areas. It has applications in image processing.

Deep results in geometric measure theory identified a dichotomy between "rectifiable" or "regular sets" and measures on the one side, and non-rectifiable or fractal sets on the other.

Some basic results in geometric measure theory can turn out to have surprisingly far-reaching consequences. For example, the Brunn-Minkowski inequality for the "n"-dimensional volumes of convex bodies "K" and "L",

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

can be proved on a single page, yet 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.

One application of geometric measure theory is the proof of Plateau's laws by Jean Taylor (building off work of Frederick J. Almgren, Jr.).

See also

* Coarea formula
* Herbert Federer
* Plateau's problem

References

* citation
last = Federer
first = Herbert
authorlink = Herbert Federer
title = Geometric measure theory
series Die Grundlehren der mathematischen Wissenschaften, Band 153
publisher = Springer-Verlag New York Inc.
location = New York
year = 1969
pages = xiv+676
ISBN = 978-3540606567
id= MathSciNet|id=0257325
*citation|first=H. |last=Federer|title=Colloquium lectures on geometric measure theory|journal= Bull. Amer. Math. Soc. |volume= 84 |issue= 3 |year=1978|pages= 291–338
url=http://www.ams.org/bull/1978-84-03/S0002-9904-1978-14462-0/
* citation
last=Gardner
first=Richard J.
title=The Brunn-Minkowski inequality
journal=Bull. Amer. Math. Soc. (N.S.)
volume=39
issue=3
year=2002
pages=355–405 (electronic)
issn = 0273-0979
id= MathSciNet|id=1898210
url= http://www.ams.org/bull/2002-39-03/S0273-0979-02-00941-2/
* citation
last = Morgan
first = Frank
title = Geometric measure theory: A beginner's guide
edition = Third edition
publisher = Academic Press Inc.
location = San Diego, CA
year = 2000
pages = x+226
isbn = 0-12-506851-4
id=MathSciNet|id=1775760
*springer|id=G/g130040|first=T.C. |last=O'Neil

External links

* Peter Mörters' GMT page [http://www.mathematik.uni-kl.de/~peter/gmt.html]
* Toby O'Neil's GMT page with references [http://mcs.open.ac.uk/tcon2/gmt.htm]


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