Geometric and stochastic representations are derived for the big class of p-generalized elliptically contoured distributions, and (generalizing Cavalieri?s and Torricelli?s method of indivisibles in a non-Euclidean sense) a geometric disintegration method is established for deriving even more general star-shaped distributions. Applications to constructing non-concentric elliptically contoured and generalized von Mises distributions are presented.

Elementary trigonometric quantities are defined in l 2,p analogously to that in l 2,2 , the sine and cosine functions are generalized for each p > 0 as functions sin p and cos p such that they satisfy the basic equation |cos p (ϕ)| p + |sin p (ϕ)| p = 1. The p-generalized radius coordinate of a point ξ ∈ R n is defined for each p > 0 as r p = ( n i=1 |ξ i | p ) 1/p . On combining these quantities, l n,p -spherical coordinates are defined. It is shown that these coordinates are nearly related to l n,p -simplicial coordinates. The Jacobians of these generalized coordinate transformations are derived. Applications and interpretations from analysis deal especially with the definition of a generalized surface content on l n,p -spheres which is nearly related to a modified co-area formula and an extension of Cavalieri's and Torricelli's indivisibeln method, and with differential equations. Applications from probability theory deal especially with a geometric interpretation of the uniform probability distribution on the l n,p -sphere and with the derivation of certain generalized statistical distributions.

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Integral representations of the locally defined star-generalized surface content measures on star spheres are derived for boundary spheres of balls being convex or radially concave with respect to a fan inRn. As a result, the general geometric measure representation of star-shaped probability distributions and the general stochastic representation of the corresponding random vectors allow additional specific interpretations in the two mentioned cases. Applications to estimating and testing hypotheses on scaling parameters are presented, and two-dimensional sample clouds are simulated.