Given two (real) normed (linear) spaces
$X$
and
$Y$
, let
$X\otimes _{1}Y=(X\otimes Y,\Vert \cdot \Vert )$
, where
$\Vert (x,y)\Vert =\Vert x\Vert +\Vert y\Vert$
. It is known that
$X\otimes _{1}Y$
is
$2$
-UR if and only if both
$X$
and
$Y$
are UR (where we use UR as an abbreviation for uniformly rotund). We prove that if
$X$
is
$m$
-dimensional and
$Y$
is
$k$
-UR, then
$X\otimes _{1}Y$
is
$(m+k)$
-UR. In the other direction, we observe that if
$X\otimes _{1}Y$
is
$k$
-UR, then both
$X$
and
$Y$
are
$(k-1)$
-UR. Given a monotone norm
$\Vert \cdot \Vert _{E}$
on
$\mathbb{R}^{2}$
, we let
$X\otimes _{E}Y=(X\otimes Y,\Vert \cdot \Vert )$
where
$\Vert (x,y)\Vert =\Vert (\Vert x\Vert _{X},\Vert y\Vert _{Y})\Vert _{E}$
. It is known that if
$X$
is uniformly rotund in every direction,
$Y$
has the weak fixed point property for nonexpansive maps (WFPP) and
$\Vert \cdot \Vert _{E}$
is strictly monotone, then
$X\otimes _{E}Y$
has WFPP. Using the notion of
$k$
-uniform rotundity relative to every
$k$
-dimensional subspace we show that this result holds with a weaker condition on
$X$
.