  In this work we study a class of random convex sets that "interpolate"
between polytopes and zonotopes. These sets arise from considering a
$q^{th}$-moment ($q\geq 1$) of an average of order statistics of
$1$-dimensional marginals of a sequence of $N\geq n$ independent random vectors
in $\mathbb R^n$. We consider the random model of isotropic log-concave
distributions as well as the uniform distribution on an $\ell_p^n$-sphere
($1\leq p < \infty$) with respect to the cone probability measure, and study
the geometry of these sets in terms of the support function and mean width. We
provide asymptotic formulas for the expectation of these geometric functionals
which are sharp up to absolute constants. Our model includes and generalizes
the standard one for random polytopes.
