# Moscow Mathematical Journal

Volume 20, Issue 1, January–March 2020 pp. 27–42.

Matrix Polar Decomposition and Generalisations of the Blaschke–Petkantschin Formula in Integral Geometry

**Authors**:
Peter J. Forrester (1)

**Author institution:**(1) Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia

**Summary: **

In the work [Bull. Austr. Math. Soc. **85** (2012), 315–234],
S. R. Moghadasi has shown how the decomposition of the $N$-fold
product of Lebesgue measure on $\mathbb{R}^n$ implied by matrix polar
decomposition can be used to derive the Blaschke–Petkantschin
decomposition of measure formula from integral geometry. We use known
formulas from random matrix theory to give a simplified derivation of
the decomposition of Lebesgue product measure implied by matrix polar
decomposition, applying too to the cases of complex and real
quaternion entries, and we give corresponding generalisations of the
Blaschke–Petkantschin formula. A number of applications to random
matrix theory and integral geometry are given, including to the
calculation of the moments of the volume content of the convex hull of
$k \le N+1$ points in $\mathbb{R}^N$, $\mathbb{C}^N$ or $\mathbb{H}^N$
with a Gaussian or uniform distribution.

2010 Math. Subj. Class. 15B52; 52A22.

**Keywords:**Blaschke–Petkantschin formula, matrix polar decomposition, integral geometry

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