  In this paper we study $L_p$-norm spherical copulas for arbitrary $p \in
[1,\infty]$ and arbitrary dimensions. The study is motivated by a conjecture
that these distributions lead to a sharp bound for the value of a certain
generalized mean difference. We fully characterize conditions for existence and
uniqueness of $L_p$-norm spherical copulas. Explicit formulas for their
densities and correlation coefficients are derived and the distribution of the
radial part is determined. Moreover, statistical inference and efficient
simulation are considered.
