  The symmetric convex hull of random points that are independent and
distributed according to the cone probability measure on the $\ell_p$-unit
sphere of $\mathbb R^n$ for some $1\leq p < \infty$ is considered. We prove
that these random polytopes have uniformly absolutely bounded isotropic
constants with overwhelming probability. This generalizes the result for the
Euclidean sphere ($p=2$) obtained by D. Alonso-Guti\'errez. The proof requires
several different tools including a probabilistic representation of the cone
measure due to G. Schechtman and J. Zinn and moment estimates for sums of
independent random variables with log-concave tails originating in the work of
E. Gluskin and S. Kwapie\'n.
