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In the work [Bull, Austr. Math. Soc. 85 (2012), 315-234], S.R. Moghadasi hasshown 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 theBlaschke-Petkantschin decomposition of measure formula from integral geometry.We use known formulas from random matrix theory to give a simplified derivationof the decomposition of Lebesgue product measure implied by matrix polardecomposition, applying too to the cases of complex and real quaternionentries, and we give corresponding generalisations of theBlaschke--Petkantschin formula. A number of applications to random matrixtheory and integral geometry are given, including to the calculation of themoments 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 uniformdistribution.

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