[For the first part of this paper see ESAIM, Control, Optimisation and Calculus of Variations.] Motivated by the study of multidimensional control problems of Dieudonné-Rashevsky type, e.g. nonconvex correspondence problems from image processing, we raise the question how to understand to notion of quasiconvexity for a continuous function
f
f
with a convex body
\mathrm{K} \subset \mathbb{R}^{nm}
K
⊂
R
n
m
instead of the whole space
\mathbb{R}^{nm}
R
n
m
as the range of definition. Extending
f
f
by
(+\infty)
(
+
∞
)
to the complement
\mathbb{R}^{nm}\setminus\,\mathrm{K}
R
n
m
∖
K
, the appropriate quasiconvex envelope turns out to be
f^{(qc)}(w) = \sup\, \bigl\{g(w)\, \big\vert\, g \colon \mathbb{R}^{nm} \to \mathbb{R} \cup \{(+\infty)\}
f
(
q
c
)
(
w
)
=
sup
{
g
(
w
)
∣
g
:
R
n
m
→
R
∪
{
(
+
∞
)
}
quasiconvex and lower semicontinuous,
g(v) \le f(v)\ \forall v \in \mathbb{R}^{nm}\bigr\}.
g
(
v
)
≤
f
(
v
)
∀
v
∈
R
n
m
}
.
In the present paper, we prove that
f^{(qc)}
f
(
q
c
)
admits a representation as
f^{(qc)}(w) =
f
(
q
c
)
(
w
)
=
Min
\bigl\{\int_{\mathrm{K}} f(v)\,d\nu(v)\, \big\vert\, \nu \in \mathrm{S}^{(qc)}(w)\bigr\} \quad \forall w \in \mathrm{K}
{
∫
K
f
(
v
)
d
ν
(
v
)
∣
ν
∈
S
(
q
c
)
(
w
)
}
∀
w
∈
K
where the sets
\mathrm{S}^{(qc)} (w)
S
(
q
c
)
(
w
)
are nonempty, convex, weak
^*
∗
-sequentially compact subsets of probability measures. This theorem, forming a natural counterpart to the author's previous results about the representation of
f^{(qc)}
f
(
q
c
)
in terms of Jacobi matrices, has been proven indispensable for the derivation of Jensens' integral inequality as well as of differentiability theorems for the envelope
f^{(qc)}
f
(
q
c
)
. The paper is mainly concerned with a detailed analysis of the set-valued map
\mathrm{S}^{(qc)}
S
(
q
c
)
, which will be explicitely described in terms of averages of generalized controls.