  We study some geometric properties of the $L_q$-centroid bodies $Z_q(\mu)$ of
an isotropic log-concave measure $\mu $ on ${\mathbb R}^n$. For any $2\ls
q\ls\sqrt{n}$ and for $\varepsilon \in (\varepsilon_0(q,n),1)$ we determine the
inradius of a random $(1-\varepsilon)n$-dimensional projection of $Z_q(\mu)$ up
to a constant depending polynomially on $\varepsilon $. Using this fact we
obtain estimates for the covering numbers
$N(\sqrt{\smash[b]{q}}B_2^n,tZ_q(\mu))$, $t\gr 1$, thus showing that $Z_q(\mu)$
is a $\beta $-regular convex body. As a consequence, we also get an upper bound
for $M(Z_q(\mu))$.
