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

Abstract

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 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 ≤ N + 1 points in R N , C N or H N with a Gaussian or uniform distribution.

Lyapunov Exponents for Some Isotropic Random Matrix Ensembles

“…The moments for the Gaussian weight for all three number systems, and similarly the moments for the beta type I weight with ν l = 0 (uniform on the sphere) and βν l /2 = 1 (uniform on the ball), have been computed in the recent work [12,Prop. 8]; see also [22] in the Gaussian case.…”

Derivation of a Formula of Blaschke and Petkantschin using Probabilistic Ideas

“…In its classical form, it can be interpreted as a decomposition of k-fold product measure of n-dimensional Euclidean space. However it has been restated and generalized by many authors; see [10,12,6,9]. Most of these works have used differential forms.…”

A note on the Blaschke-Petkantschin formula, Riesz distributions, and Drury’s identity

“…After finishing the first version of the paper, the author became aware of close works by Moghadasi [19] and Forrester [10] devoted to application of the matrix polar decomposition to derivation of the Blaschke-Petkantschin formula. Our reasoning essentially differs from [10,19].…”