- Geometric measure theory
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
fractalsets on the other.
Some basic results in geometric measure theory can turn out to have surprisingly far-reaching consequences. For example, the
Brunn-Minkowski inequalityfor the "n"-dimensional volumes of convex bodies "K" and "L",
can be proved on a single page, yet quickly yields the classical
isoperimetric inequality. The Brunn-Minkowski inequality also leads to Anderson's theoremin statistics. The proof of the Brunn-Minkowski inequality predates modern measure theory; the development of measure theory and Lebesgue integrationallowed 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 inequalitythe geometry seems almost entirely absent.
One application of geometric measure theory is the proof of
Plateau's lawsby Jean Taylor(building off work of Frederick J. Almgren, Jr.).
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
*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
title=The Brunn-Minkowski inequality
journal=Bull. Amer. Math. Soc. (N.S.)
issn = 0273-0979
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
* 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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