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

  • S. REZA MOGHADASI (a1)

Abstract

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.

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References

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[1]Evans, L. C. and Gariepy, R. F., Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics (CRC Press, Boca Raton, FL, 1999).
[2]Federer, H., Geometric Measure Theory (Springer, Berlin, 1996).
[3]Mehta, M. L., Random Matrices (Elsevier/Academic Press, Amsterdam, 2004).
[4]Vedel Jensen, E. B., Local Stereology (World Scientific, River Edge, NJ, 1998).
[5]Vedel Jensen, E. B. and Kiêu, K., ‘A new integral geometric formula of the Blaschke–Petkantschin type’, Math. Nachr. 156 (1992), 5774.
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POLAR DECOMPOSITION OF THE k-FOLD PRODUCT OF LEBESGUE MEASURE ON ℝn

  • S. REZA MOGHADASI (a1)

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