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GEOMETRIC AND FIXED POINT PROPERTIES IN PRODUCTS OF NORMED SPACES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 99 / Issue 2 / April 2019
- Published online by Cambridge University Press:
- 10 January 2019, pp. 262-273
- Print publication:
- April 2019
-
- Article
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- You have access
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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 
$X$.
