  A probabilistic representation for a class of weighted $p$-radial
distributions, based on mixtures of a weighted cone probability measure and a
weighted uniform distribution on the Euclidean $\ell_p^n$-ball, is derived.
Large deviation principles for the empirical measure of the coordinates of
random vectors on the $\ell_p^n$-ball with distribution from this weighted
measure class are discussed. The class of $p$-radial distributions is extended
to $p$-balls in classical matrix spaces, both for self-adjoint and
non-self-adjoint matrices. The eigenvalue distribution of a self-adjoint random
matrix, chosen in the matrix $p$-ball according to such a distribution, is
determined. Similarly, the singular value distribution is identified in the
non-self-adjoint case. Again, large deviation principles for the empirical
spectral measures for the eigenvalues and the singular values are presented as
an application.
