Let
K
K
be a convex body in
R
d
\mathbb {R}^d
, let
j
∈
{
1
,
…
,
d
−
1
}
j\in \{1, \ldots , d-1\}
, and let
ϱ
\varrho
be a positive and continuous probability density function with respect to the
(
d
−
1
)
(d-1)
-dimensional Hausdorff measure on the boundary
∂
K
\partial K
of
K
K
. Denote by
K
n
K_n
the convex hull of
n
n
points chosen randomly and independently from
∂
K
\partial K
according to the probability distribution determined by
ϱ
\varrho
. For the case when
∂
K
\partial K
is a
C
2
C^2
submanifold of
R
d
\mathbb {R}^d
with everywhere positive Gauss curvature, M. Reitzner proved an asymptotic formula for the expectation of the difference of the
j
j
th intrinsic volumes of
K
K
and
K
n
K_n
, as
n
→
∞
n\to \infty
. In this article, we extend this result to the case when the only condition on
K
K
is that a ball rolls freely in
K
K
.