Let
X
1
{X_1}
and
X
2
{X_2}
be Banach spaces, and let
X
1
×
X
2
{X_1} \times {X_2}
be equipped with the
l
1
{l_1}
-norm. If the first space
X
1
{X_1}
is uniformly convex in every direction, then
X
1
×
X
2
{X_1} \times {X_2}
has the fixed point property for nonexpansive mappings (FPP) if and only if
R
×
X
2
\mathbb {R} \times {X_2}
(with the
l
1
{l_1}
-norm) does. If
X
1
{X_1}
is merely strictly convex,
(
R
×
X
2
)
(\mathbb {R} \times {X_2})
has the FPP, and
C
i
⊂
X
i
{C_i} \subset {X_i}
are weakly compact and convex with the FPP (for
i
=
1
,
2
i = 1,2
), then the fixed point set of every nonexpansive mapping
T
:
C
1
×
C
2
→
C
1
×
C
2
T:{C_1} \times {C_2} \to {C_1} \times {C_2}
is a nonexpansive retract of
C
1
×
C
2
{C_1} \times {C_2}
.