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POLAR DECOMPOSITION OF THE k-FOLD PRODUCT OF LEBESGUE MEASURE ON ℝn

Part of: Geometric probability and stochastic geometry Classical measure theory

Published online by Cambridge University Press:  06 January 2012

S. REZA MOGHADASI
S. REZA MOGHADASI*
Affiliation:
Department of Mathematical Science, Sharif University of Technology, PO Box 11155-9415, Tehran, Iran (email: [email protected])
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Abstract

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The Blaschke–Petkantschin formula is a geometric measure decomposition of the q-fold product of Lebesgue measure on ℝn. Here we discuss another decomposition called polar decomposition by considering ℝn×⋯×ℝn as ℳn×k and using its polar decomposition. This is a generalisation of the Blaschke–Petkantschin formula and may be useful when one needs to integrate a function g:ℝn×⋯×ℝn→ℝ with rotational symmetry, that is, for each orthogonal transformation O,g(O(x1),…,O(xk))=g(x1,…xk). As an application we compute the moments of a Gaussian determinant.

Keywords

Blaschke–Petkantschin formulaco-area formulapolar decompositionmoments of Gaussian determinant

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory 60D05: Geometric probability and stochastic geometry
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 85 , Issue 2 , April 2012 , pp. 315 - 324
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

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