Assume that
X
X
is a Banach space of measurable functions for which Komlós’ Theorem holds. We associate to any closed convex bounded subset
C
C
of
X
X
a coefficient
t
(
C
)
t(C)
which attains its minimum value when
C
C
is closed for the topology of convergence in measure and we prove some fixed point results for affine Lipschitzian mappings, depending on the value of
t
(
C
)
∈
[
1
,
2
]
t(C)\in [1,2]
and the value of the Lipschitz constants of the iterates. As a first consequence, for every
L
>
2
L>2
, we deduce the existence of fixed points for affine uniformly
L
L
-Lipschitzian mappings defined on the closed unit ball of
L
1
[
0
,
1
]
L_1[0,1]
. Our main theorem also provides a wide collection of convex closed bounded sets in
L
1
(
[
0
,
1
]
)
L^1([0,1])
and in some other spaces of functions which satisfy the fixed point property for affine nonexpansive mappings. Furthermore, this property is still preserved by equivalent renormings when the Banach-Mazur distance is small enough. In particular, we prove that the failure of the fixed point property for affine nonexpansive mappings in
L
1
(
μ
)
L_1(\mu )
can only occur in the extremal case
t
(
C
)
=
2
t(C)=2
. Examples are given proving that our fixed point theorem is optimal in terms of the Lipschitz constants and the coefficient
t
(
C
)
t(C)
.