  Central limit theorems for the log-volume of a class of random convex bodies
in $\mathbb{R}^n$ are obtained in the high-dimensional regime, that is, as
$n\to\infty$. In particular, the case of random simplices pinned at the origin
and simplices where all vertices are generated at random is investigated. The
coordinates of the generating vectors are assumed to be independent and
identically distributed with subexponential tails. In addition, asymptotic
normality is established also for random convex bodies (including random
simplices pinned at the origin) when the spanning vectors are distributed
according to a radially symmetric probability measure on the $n$-dimensional
$\ell_p$-ball. In particular, this includes the cone and the uniform
probability measure.
