  The geometry of unit $N$-dimensional $\ell_{p}$ balls has been intensively
investigated in the past decades. A particular topic of interest has been the
study of the asymptotics of their projections. Apart from their intrinsic
interest, such questions have applications in several probabilistic and
geometric contexts (Barthe et al. 2005). In this paper, our aim is to revisit
some known results of this flavour with a new point of view. Roughly speaking,
we will endow the ball with some kind of Dirichlet distribution that
generalizes the uniform one and will follow the method developed in Skibinsky
(1967), Chang et al. (1993) in the context of the randomized moment space. The
main idea is to build a suitable coordinate change involving independent random
variables. Moreover, we will shed light on a nice connection between the
randomized balls and the randomized moment space.
