Matrix polar decomposition and generalisations of the Blaschke–Petkantschin formula in integral geometry

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.