Let
C
C
and
K
K
be centrally symmetric convex bodies in
R
n
{\mathbb R}^n
. We show that if
C
C
is isotropic then
‖
t
‖
C
s
,
K
=
∫
C
⋯
∫
C
‖
∑
j
=
1
s
t
j
x
j
‖
K
d
x
s
⋯
d
x
1
⩽
c
1
L
C
(
log
n
)
5
n
M
(
K
)
‖
t
‖
2
\begin{equation*} \|\mathbf {t}\|_{C^s,K}=\int _{C}\cdots \int _{C}\Big \|\sum _{j=1}^st_jx_j\Big \|_K\,dx_s\cdots dx_1 \leqslant c_1L_C(\log n)^5\,\sqrt {n}M(K)\|\mathbf {t}\|_2 \end{equation*}
for every
s
⩾
1
s\geqslant 1
and
t
=
(
t
1
,
…
,
t
s
)
∈
R
s
\mathbf {t}=(t_1,\ldots ,t_s)\in {\mathbb R}^s
, where
L
C
L_C
is the isotropic constant of
C
C
and
M
(
K
)
≔
∫
S
n
−
1
‖
ξ
‖
K
d
σ
(
ξ
)
M(K)≔\int _{S^{n-1}}\|\xi \|_Kd\sigma (\xi )
. This reduces a question of V. Milman to the problem of estimating from above the parameter
M
(
K
)
M(K)
of an isotropic convex body. The proof is based on an observation that combines results of Eldan, Lehec and Klartag on the slicing problem: If
μ
\mu
is an isotropic log-concave probability measure on
R
n
{\mathbb R}^n
then, for any centrally symmetric convex body
K
K
in
R
n
{\mathbb R}^n
, we have that
I
1
(
μ
,
K
)
≔
∫
R
n
‖
x
‖
K
d
μ
(
x
)
⩽
c
2
n
(
log
n
)
5
M
(
K
)
.
\begin{equation*} I_1(\mu ,K)≔\int _{{\mathbb R}^n}\|x\|_K\,d\mu (x)\leqslant c_2\sqrt {n}(\log n)^5\,M(K). \end{equation*}
We illustrate the use of this inequality with further applications.