LAUSR

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 $X$ .