Motivated by the separability problem in quantum systems 2⊗4, 3⊗3 and 2⊗2⊗2, we study the maximal (proper) faces of the convex body,
S
1
, of normalized separable states in an arbitrary quantum system with finite-dimensional Hilbert space
H
=
H
1
⊗
H
2
⊗
⋯
⊗
H
n
. To any subspace
V
⊆
H
, we associate a face
F
V
of
S
1
consisting of all states
ρ
∈
S
1
whose range is contained in
V
. We prove that
F
V
is a maximal face if and only if
V
is a hyperplane. If
V
=|
ψ
〉
⊥
, where |
ψ
〉 is a product vector, we prove that
Dim
F
V
=
d
2
−
1
−
∏
(
2
d
i
−
1
)
, where
d
i
=
Dim
H
i
and
d
=
∏
d
i
. We classify the maximal faces of
S
1
in the cases 2⊗2 and 2⊗3. In particular, we show that the minimum and the maximum dimension of maximal faces is 6 and 8 for 2⊗2, and 20 and 24 for 2⊗3. The boundary,
∂
S
1
, of
S
1
is the union of all maximal faces. When
d
>6, it is easy to show that there exist full states on
∂
S
1
, i.e. states
ρ
∈
∂
S
1
such that all partial transposes of
ρ
(including
ρ
itself) have rank
d
. Ha and Kye have recently constructed explicit such states in 2⊗4 and 3⊗3. In the latter case, they have also constructed a remarkable family of faces, depending on a real parameter
b
>0,
b
≠1. Each face in the family is a nine-dimensional simplex, and any interior point of the face is a full state. We construct suitable optimal entanglement witnesses for these faces and analyse the three limiting cases
b
=
0
,
1
,
∞
.