- Geometric measure theory
In

mathematics ,**geometric measure theory**(**GMT**) is the study of the geometric properties of the measures of sets (typically inEuclidean space s), including such things asarc length s andarea s. It has applications inimage 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 toAnderson's theorem in statistics. The proof of the Brunn-Minkowski inequality predates modern measure theory; the development of measure theory andLebesgue 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 thePrékopa-Leindler inequality the geometry seems almost entirely absent.One application of geometric measure theory is the proof of

Plateau's laws byJean Taylor (building off work ofFrederick 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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