We study some geometric properties of the
L
q
L_q
-centroid bodies
Z
q
(
μ
)
Z_q(\mu )
of an isotropic log-concave measure
μ
\mu
on
R
n
{\mathbb R}^n
. For any
2
⩽
q
⩽
n
2\leqslant q\leqslant \sqrt {n}
and for
ε
∈
(
ε
0
(
q
,
n
)
,
1
)
\varepsilon \in (\varepsilon _0(q,n),1)
we determine the inradius of a random
(
1
−
ε
)
n
(1-\varepsilon )n
-dimensional projection of
Z
q
(
μ
)
Z_q(\mu )
up to a constant depending polynomially on
ε
\varepsilon
. Using this fact we obtain estimates for the covering numbers
N
(
[
b
]
q
B
2
n
,
t
Z
q
(
μ
)
)
N(\sqrt {[b]{q}}B_2^n,tZ_q(\mu ))
,
t
⩾
1
t\geqslant 1
, thus showing that
Z
q
(
μ
)
Z_q(\mu )
is a
β
\beta
-regular convex body. As a consequence, we also get an upper bound for
M
(
Z
q
(
μ
)
)
M(Z_q(\mu ))
.