  For an isotropic convex body $K\subset\mathbb{R}^n$ we consider the isotropic
constant $L_{K_N}$ of the symmetric random polytope $K_N$ generated by $N$
independent random points which are distributed according to the cone
probability measure on the boundary of $K$. We show that with overwhelming
probability $L_{K_N}\leq C\sqrt{\log(2N/n)}$, where $C\in(0,\infty)$ is an
absolute constant. If $K$ is unconditional we argue that even $L_{K_N}\leq C$
with overwhelming probability. The proofs are based on concentration
inequalities for sums of sub-exponential or sub-Gaussian random variables,
respectively, and, in the unconditional case, on a new $\psi_2$-estimate for
linear functionals with respect to the cone measure in the spirit of Bobkov and
Nazarov, which might be of independent interest.
