ISOTROPIC RANDOM SIMPLICES

Some obscure, yet fundamental, formulae of integral geometry are re-considered. They are applied to determine all the moments of the random volume of various isotropic random r-dimensional simplices in En (r = 1,...,n). This paper has two main objects. First, to draw attention to certain little known, yet beautiful and fundamental, 'Jacobian' formulae of integral geometry, due to Blaschke and his school in the 1930's; this may be especially helpful for English-speaking probabilists and statisticians. Moreover, to give simple (somewhat heuristic) derivations of these formulae by means of stochastic-type 'conditional' arguments, thereby revealing their intrinsic nature. Second, to apply them to certain interesting problems in random geometry, while incidentally noting intriguing connections with the Wishart multivariate normal theory. Several of the results were presented by the author at the

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