Geometric representations of multivariate skewed elliptically contoured distributions

We derive a wide class of geometric representation formulas for multivariate skewed elliptically contoured distributions and show in a unified geometric way how some of them are related to stochastic representations known in the literature. Furthermore, we make use of the geometric measure representation to explore independence between collections of components of accordingly distributed random vectors, and to investigate contour plots of skewed normal densities from a geometric viewpoint.