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\topmatter
\title=20
Lambda-Hyperconvexity in Metric Spaces
\endtitle
\rightheadtext{LAMBDA-HYPERCONVEXITY IN METRIC SPACES}

\author M.A. Khamsi, H. Knaust, N.T. Nguyen and M.D. O'Neill \endauthor
\address Department of Mathematical Sciences, The University of Texas at El=
 Paso, El Paso, TX~79968-0514, USA\endaddress
\email mohamed\@math.utep.edu\endemail
\address Department of Mathematical Sciences, The University of Texas at El=
 Paso, El Paso, TX~79968-0514, USA\endaddress
\email helmut\@math.utep.edu\endemail
\address Department of Mathematical Sciences, The University of Texas at El=
 Paso, El Paso, TX~79968-0514, USA\endaddress
\email nguyen\@math.utep.edu\endemail
\address Department of Mathematical Sciences, The University of Texas at El=
 Paso, El Paso, TX~79968-0514, USA\endaddress
\email michael\@math.utep.edu\endemail

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\abstract=20
We introduce the concept of $\lambda$-hyperconvexity in metric spaces,=
 generalizing the classical notion of a hyperconvex metric space.  We show=
 that a bounded metric space which is $\lambda$-hyperconvex has the fixed=
 point property for nonexpansive mappings provided $\lambda < 2$.  Uniformly=
 convex Banach spaces are examples of such $\lambda$-hyperconvex spaces for=
 some $\lambda < 2$. We furthermore investigate the relationship between=
 Penot's Intersection Property and $2$-hyperconvexity.
\endabstract
\keywords
Hyperconvexity, fixed point
\endkeywords
\subjclass  Primary
46B20, 47E10\endsubjclass
\endtopmatter

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\heading{\S 1. Introduction}\endheading
Aronszajn and Panitchpakdi \cite{AP} define a metric space to be {\it=
 hyperconvex} if it satisfies the following condition:  Whenever a=
 collection of balls intersects pairwise, the intersection of all balls in=
 the collection is not empty. We generalize this concept by calling  a=
 metric space {\it $\lambda$-hyperconvex} if {\it every pairwise=
 intersecting collection of balls with centers in a given admissible set has=
 non-empty intersection in the admissible set if the radii of the balls are=
 increased by  the factor $\lambda$} (see Definition~2 below for a precise=
 definition). While the Hilbert space $\ell_2$ fails to be hyperconvex, it=
 naturally satisfies the condition of $\lambda$-hyperconvexity for =
 $\lambda=3D\sqrt{2}$.=20

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The
recent interest into hyperconvex spaces goes back to  results of Sine
\cite{Si2} and Soardi \cite{Soa} who proved independently that the fixed=
 point property for nonexpansive mappings holds in bounded hyperconvex=
 spaces.  Since then many interesting results \cite{Ba, KLS, KR, LS, Si1}=
 have been
shown to hold in hyperconvex spaces. We show in Section~2 that for=
 $\lambda<2$ the fixed point property holds also for nonexpansive mappings=
 in bounded $\lambda$-hyperconvex  spaces.=20

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In Section~3 we show that uniformly convex Banach  spaces are=
 $\lambda$-hyperconvex for some $\lambda<2$, while in Section~4 we=
 investigate the relationship between Penot's Intersection Property and=
 $2$-hyperconvexity.

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\definition{Definition~1} A metric space $(M,d)$ is {\it hyperconvex} if and=
 only if for
any family of points $(x_\alpha)_{\alpha \in \Lambda}$ and any family of=
 positive
numbers $(r_\alpha)_{\alpha \in \Lambda}$ such that
$d(x_\alpha,x_\beta) \leq r_\alpha + r_\beta \text{ for all } \alpha,\beta=
 \in \Lambda$
we must have
$$\bigcap_{\alpha \in \Lambda} B(x_\alpha,r_\alpha) \neq \emptyset, $$
where $B(x,r)$ denotes the closed ball centered at $x \in M$ with
radius $r$.=20
\enddefinition
$\ell_\infty(I)$ for any index set $I$ is the classical example of a=
 hyperconvex metric space, while the Hilbert space $\ell_2$ fails to be=
 hyperconvex.

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We may refine this definition by considering the cardinality of the
index set $\Lambda$.  We will say that $(M,d)$ (or $M$ for
short) is {\it finitely hyperconvex} (resp. {\it sequentially hyperconvex})=
 if the
property above is satisfied for any finite set $\Lambda$ (resp. countable
set $\Lambda$). =20

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Recall that a subset $A$ of $M$ is called {\it admissible} if and only
if $A$ is an intersection of closed balls.  The family of admissible
subsets of $M$ will be  denoted by $\Cal{A}(M)$.

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This article is centered around the following notion:
\definition{Definition 2} Let $M$ be a metric space and
let $\lambda\geq 1$.  We say that the metric space $M$ is {\it
$\lambda$-hyperconvex} if for every non-empty admissible set $A$ of
$M$, for any family of closed balls
$\{B(x_{\alpha},r_{\alpha})\}_{\alpha\in\Lambda}$, each of radius=20
$r_{\alpha}$, centered at $x_{\alpha}\in A$ for $\alpha\in\Lambda$, the
condition
$d(x_{\alpha},x_{\beta})\leq r_{\alpha}+r_{\beta} \text{ for every }\alpha,
\beta\in \Lambda,$=20
implies=20
$$A\cap \Big(\bigcap_{\alpha\in\Lambda} B(x_{\alpha},\lambda=
 r_{\alpha})\Big)\neq
\emptyset.$$
We let $\Lambda(M)$ be the infimum of all constants $\lambda$ such that $M$=
 is $\lambda$-hyperconvex, and say that $\Lambda(M)$ is {\it exact} if $M$=
 is $\Lambda(M)$-hyperconvex.
\enddefinition

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Gr\"unbaum~\cite{Gr1,Gr2,Gr3} and other authors have studied a similar=
 property not involving the
underlying admissible set $A$.  For our purposes it is essential to include=
 the restriction that the intersection of the expanded balls also intersects=
 with the admissible set containing the centers of the original balls.=
 Otherwise we will not be able to connect this concept to the normal=
 structure property (see the proof
of Theorem~2 for more details).

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Let us recall Gr\"unbaum's definition: For a metric space $M$, let the {\it=
 expansion constant} $E(M)$ be the infimum of all constants $\mu$ such that=
 the following holds: Whenever a collection $\{B(x_\alpha,r_\alpha) :\=
 \alpha\in \Lambda\}$ intersects pairwise, then
$$ \bigcap_{\alpha\in\Lambda}B(x_\alpha,\mu\cdot r_\alpha)\not=3D\emptyset.$=
$
We say $E(M)$ is {\it exact}, if the condition is even satisfied for=
 $\mu=3DE(M)$. =20

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Trivially, $E(M)\leq\Lambda(M)$ holds in metrically convex spaces. On the=
 other hand, if $M$ is a two element metric space, then $E(M)=3D1$, while=
 $\Lambda(M)=3D2$, so both concepts do not coincide in general.

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Let us first summarize some basic properties of $\lambda$-hyperconvex metric=
 spaces, some of which are trivial, while the others can be easily derived=
 from corresponding results about expansion constants:=20

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\proclaim{Theorem~1} \roster
\runinitem A metric space is hyperconvex if and only if it is=
 $1$-hyperconvex.
\item For every complete metric space $M$, $\Lambda(M)\leq 2$~\cite{Gr2}.=
 Every $\lambda$-hyperconvex metric space is complete. =20
\item   If $\Cal{A}(M)$ has the Intersection Property, then $\Lambda(M)$ is=
 exact~\cite{Gr2, Theorem~3}.
Here we say, $\Cal{A}(M)$ has the {\it Intersection Property} if and only if=
 any family of elements $\{A_{\alpha}\}_{\alpha
\in \Lambda}$ in $\Cal{A}(M)$ has a nonempty intersection provided=20
$$\bigcap_{\alpha\in \Lambda_f} A_\alpha\neq \emptyset$$
for any finite subset $\Lambda_f$ of $\Lambda$~\cite{Pe}.
\item In particular, reflexive Banach spaces and dual Banach spaces are=
 $2$-hyperconvex.=20
\item There is a subspace $X$ of $\ell_1$, which fails to be=
 $2$-hyperconvex~\cite{Gr2}.
 \item Hilbert space is $\sqrt{2}$-hyperconvex~\cite{Ju, Gr1}.
\endroster\endproclaim
\remark{Remark}The counterexample in (5) is the following example, first=
 considered by Klee~\cite{Kl} in a different context:
$$X=3D\left\{(x_n)_{n\in\Bbb N} \in \ell_1:\ \sum_{n=3D1}^\infty=
 \frac{n}{n+1} x_n=3D0\right\}.$$
\endremark

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\heading{\S 2. $\lambda$-Hyperconvexity and the Fixed Point=
 Property}\endheading
We establish the following generalization of the Fixed Point Theorem by=
 Sine~\cite{Si2} and Soardi~\cite{Soa}:

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\proclaim{Theorem 2}  Let $M$ be a bounded $\lambda$-hyperconvex space. If
$\lambda<2$, then
any nonexpansive map $f:M\rightarrow M$ has a fixed point.
\endproclaim
A map $f:M\rightarrow M$ is {\it  nonexpansive} if $$
d(f(x),f(y))\leq d(x,y) \text{ for every } x, y \in M.$$


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The proof of Theorem 2 makes use of the following notation:  Let $A$ be  a=
 bounded subset of a metric  space $M$. We set=20
$$\align=20
R(x,A)&:=3D\sup\{d(x,y): y\in A\} \text{ for } x\in M, \\
\delta(A)&:=3D\sup\{R(x,A): x\in A\}  \\
\text{and}\quad R(A)&:=3D\inf\{R(x,A): x\in A\}
\endalign$$
We also need the following definitions:
Let $\Cal F$ be a family of subsets in a
metric  space $M$. We say that $\Cal F$ defines a  {\it convexity structure=
 on $M$}
if it contains the balls and  is stable  by  intersection.

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Assume that $M$ is  a
metric  space with a convexity structure $\Cal F$. We say that
$\Cal F$ is  a {\it uniform
normal structure on $M$}
if there exists $c<1$ such that=20
$$R(A)\leq c\,\delta(A) \text{ for every } A\in \Cal F.$$=20

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The proof of Theorem 2 is based on the following fact established=
 in~\cite{Kh}.
\proclaim{Theorem} Let $M$ be a bounded complete metric space.
If $M$ has a uniform normal structure, then $M$ has the fixed point property
for nonexpansive maps.
\endproclaim

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\demo{Proof of Theorem 2} Let $M$ be a bounded
$\lambda$-hyperconvex space, with  $\lambda<2$. By a previous remark, $M$ is=
 complete. By Khamsi's result it
suffices to prove that $M$ has a uniform normal structure.=20

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 The family of admissible sets $\Cal{A}(M)$
defines a convexity structure on $M$.

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We shall   show that $\Cal{A}(M)$ is a uniform  normal structure on
$M$. Let $A\in \Cal A(M)$ with $\delta(A)>0$.
For every $x\in A$, let $B(x,r_x)$ denote the ball
centered at $x$ with constant radius $r_x=3D\frac{1}{2}\delta(A)$. Then we=
 have=20
$$d(x,y)\leq \delta(A)=3D
\frac{1}{2}\delta(A)+\frac{1}{2}\delta(A)=3Dr_x+r_y\text{ for every } x,
y \in A.$$
Since $M$ is  $\lambda$-hyperconvex  we can find =20
$$x_0\in A\cap\bigcap_{\alpha\in\Lambda} B(x_{\alpha},\lambda
r_{\alpha}).$$
Thus we have
$$d(x,x_0)\leq\frac{1}{2}\lambda\,\delta(A)\text{ for every } x \in A.
$$
It follows that=20
$$R(A)\leq\frac{1}{2}\lambda\,\delta(A).$$
Since $\frac{1}{2}\lambda<1$, $\Cal{A}(M)$ is a uniform normal structure on=
 $M$ and the
theorem is proved.
\qed\enddemo

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\remark{Remark}Alspach~\cite{Al} (see also   Lim~\cite{Lim}) constructed an=
=20
example of a weakly compact convex set in $L^1[0,1]$ which fails the fixed
point property for an isometry. From Theorem~1 we can conclude
that this example is $2$-hyperconvex,  while Theorem~2 shows that this=
 example is not $\lambda$-hyperconvex for any $\lambda<2$. In particular,=
 Theorem~2 fails for $\lambda=3D2$.\medskip
\endremark

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\heading{\S 3. Uniformly Convex Spaces}\endheading
In this section, we prove the following result:
\proclaim{Theorem 3} A uniformly convex Banach space
is $\lambda$-hyperconvex for some $\lambda < 2$.
\endproclaim

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\demo{Proof} Let $X$ be a uniformly convex Banach
space.  Since $X$ is reflexive, it is enough to show that for any
admissible subset $A$, any choice of points $x_1, \ldots, x_n \in A$ and
positive numbers $r_1, \ldots, r_n$ such that
$$d(x_i,x_j) \leq r_i + r_j\;\;\text{for all $i,j =3D 1, \ldots, n$},$$
we have
$$A \cap \Big(\bigcap_{1 \leq i \leq n} B(x_i,\lambda
r_i)\Big) \neq \emptyset$$
for some $\lambda < 2$ independent of the set $A$ and the points
$\{x_i\}$ as well as the radii $\{r_i\}$.  Without loss of
generality, we may assume that $r_1 \leq r_2 \leq \ldots \leq r_n$.
We know from the proof of Part (3) of Theorem~1 that=20
$$x_1 \in \bigcap_{1 \leq i \leq n} B(x_i,2\,r_i)\;.$$
We may assume that there exists $i\in\{1,\ldots,n\}$ such that $d(x_1, x_i)
> r_1$.  Otherwise, we have
$$x_1 \in \bigcap_{1 \leq i \leq n} B(x_i,r_1)  \subset \bigcap_{1
\leq i \leq n} B(x_i,r_i) \subset \bigcap_{1 \leq i \leq n}
B(x_i,\lambda r_i)$$
for any $\lambda \geq 1$.  Set $\Lambda =3D \{i:\; d(x_1, x_i) > r_1\}$.
Let $i_0$ be the smallest element of $\Lambda$.  Then we know that=20
$$x_{i_0} \in \bigcap_{j \in \Lambda} B(x_j,2\,r_j)$$
Since $d(x_1, x_{i_0}) > r_1$, then there exists $x_* \in
\text{Segment}[x_1, x_{i_0}]$ such that $d(x_1, x_*) =3D r_1$.  For any
$i \not\in \Lambda$, then we have=20
$$d(x_*,x_i) \leq d(x_*,x_1) + d(x_1,x_i) \leq 2r_1 \leq 2r_i \;.$$
This clearly implies
$$x_* \in \Big(\bigcap_{1 \leq i \leq n} B(x_i,2\,r_i)\Big) \cap=
 \operatorname{conv}(x_i)$$
where $\operatorname{conv}(x_i)$ is the convex hull of the points
$x_1,\ldots,x_n$.  Let $\varepsilon > 0$.  Let $i\in\{1,\ldots,n\}$.  We=
 consider
two cases:

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\noindent $\underline{\text{\bf Case 1:}}\  r_1 < \varepsilon\; r_i\;.$

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\noindent
Then we have
$$d\left(x_i, \frac{x_1+x_*}{2} \right) \leq d(x_i,x_1) + d\left(x_1,
\frac{x_1+x_*}{2} \right) \leq r_i + r_1 + \frac{r_1}{2} =3D \frac{2r_i
+ 3r_1}{2}$$
Using the above condition on $r_1$ and $r_i$, we get
$$d\left(x_i, \frac{x_1+x_*}{2} \right) \leq \frac{2+=
 3\varepsilon}{2}\;r_i$$

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\noindent $\underline{\text{\bf Case 2:}}\   \varepsilon\; r_i \leq r_1\;.$

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\noindent Since
$$d(x_1,x_*) =3D r_1 \geq 2\,r_i \frac{\varepsilon}{2}$$
and
$$d(x_i,x_1) \leq 2r_i \;\;\text{as well as}\;\;d(x_i,x_*) \leq 2r_i\;,$$=20
we get from the definition of the modulus of uniform convexity
$$d\left(x_i, \frac{x_1+x_*}{2} \right) \leq 2\,r_i \Big(1 - \delta
\left({\varepsilon}/{2}\right)\Big)\;.$$ \smallskip

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\noindent Set
$$\lambda =3D 2\, \min_{\varepsilon > 0}\left\{\max\left\{\frac{2+=
 3\varepsilon}{4},1 - \delta
\left({\varepsilon}/{2}\right)\right\}\right\}$$
Then we have
$$\frac{x_1+x_*}{2} \in=20
\Big(\bigcap_{1 \leq i \leq n} B(x_i,\lambda\, r_i)\Big) \cap
\operatorname{conv}(x_i) \subset A \cap \Big(\bigcap_{1 \leq i \leq n}=
 B(x_i,\lambda\,
r_i)\Big),$$
because $\operatorname{conv}(x_i) \subset A$.  Let us show that $\lambda <=
 2$.
Indeed, for $\varepsilon < 2/3$, we have=20
$$\max\left\{\frac{2+ 3\varepsilon}{4},1 - \delta
\left({\varepsilon}/{2}\right)\right\} < 1$$
which clearly implies $\lambda < 2$.  The proof of our claim is
therefore complete.
\qed\enddemo

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\remark{Remarks} =20
A look at the proof reveals that one  does not need the assumption that $X$=
 is uniformly convex.  It is enough to assume that the {\it characteristic=
 of uniform convexity of
$X$} is less than 1/3, that is
$$\varepsilon_0(X):=3D \sup \{\varepsilon \ :\  \delta(\varepsilon) =3D=
 0\}\; < \frac{1}{3} \;.$$
We do not know whether the above result still holds if we only assume that=
=20
$\varepsilon_0(X) < 1$.
\endremark

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\heading{\S 4. Hyperconvexity and the Intersection Property}\endheading
Clearly if $M$ is  a hyperconvex space, then $\Cal{A}(M)$ has the=
 Intersection Property. Kirk~\cite{Ki} asks whether this is still true in=
 the case of $2$-hyperconvexity. We will show that $c_0$ provides a=
 counterexample to this question.

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\proclaim{Theorem 4}=20
\roster
\runinitem $c_0$ fails the Intersection Property.
\item $c_0$ fails to be hyperconvex.=20
\item $c_0$ is finitely hyperconvex.
\item $c_0$ is $2$-hyperconvex.
\endroster\endproclaim

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\demo{Proof} Note that (1) is a direct consequence of  (2) and (3).
It is easy to find an example which shows that $c_0$ fails to be=
 hyperconvex, establishing  (2).

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To prove (3), let
$x_1,\ldots,x_K$ be elements in $c_0$ and $r_1,\ldots r_K$ be positive=
 numbers such
that
$$\|x_i - x_j \| \leq r_i + r_j$$
for $i,j =3D 1,2, \ldots, K$.  We may assume that $0 < r_1 \leq r_2 \leq
\cdots \leq r_K$.  If we write $x_i =3D (x_i(n))$, then there exists $n_0
\geq 1$ such that $|x_i(n)| \leq r_1$ for all $i=3D1,\ldots,K$, and all $n=
 \geq
n_0$.  Let $x \in c_0$ defined by=20
$$x(n) \in \Big[\max_{1 \leq i \leq n_0} (x_i - r_i)\;,\;\min_{1 \leq
i \leq n_0} (x_i + r_i)\Big]\text{ for $1\leq n\leq n_0$,}$$ and=
 $$x(n)=3D0\text{ for $n >n_0$.}$$
Clearly, each of the intervals used in the definition is non-empty, and
$$x \in \bigcap_{1 \leq i \leq K} B(x_i,r_i)\;,$$ establishing (3).

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It remains to prove that $c_0$ is $2$-hyperconvex.
Let $A$ be an admissible set in $c_0$, and let $A_\infty$ denote the=
 intersection of the same balls in $\ell_\infty$, {\it i.e.,}=
 $A=3DA_\infty\cap c_0$.
Let a collection of balls $\{B(x_\alpha,r_\alpha)\}_{\alpha\in\Lambda}$ be=
 given, each centered at some $x_\alpha\in A$ with positive radius=
 $r_\alpha$. We assume that the collection satisfies the conditions
$$\|x_\alpha-x_\beta\|\leq r_\alpha + r_\beta\text{ for all=
 }\alpha,\beta\in\Lambda.$$

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\noindent  Since $\ell_\infty$ is hyperconvex, we can find $y=3D(y(n))\in=
 A_\infty$
such that=20
$$\|y-x_\alpha\|\leq r_\alpha \text{ for all } \alpha\in\Lambda.$$=20
If we write $a=3D(a(n))$ for an element in $A_\infty$, then=20
$$\{a(n)\ :\ a\in A_\infty\}$$ is a closed interval on the real line for all=
 $n\in\Bbb N$. Thus we may assume that=20
$$\inf_{\alpha\in\Lambda}x_\alpha(n)\leq y(n)\leq=
 \sup_{\alpha\in\Lambda}x_\alpha(n)\text{ for all } n\in\Bbb N.$$
\noindent Set $r=3D\inf_{\alpha\in\Lambda} r_\alpha$.  Note that for every=
 $\varepsilon>0$, there is an $N\in\Bbb N$ such that for all $n\geq N$
$$|y(n)|\leq r+\varepsilon.$$
Indeed, pick ${\alpha_0}\in\Lambda$ such that $r_{\alpha_0}<r+\varepsilon$ =
 and observe that $$\lim_{n\to\infty}x_{\alpha_0}(n)=3D0.$$

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Next we define $x=3D(x(n))\in\ell_\infty$ as follows:
$$x(n):=3D\cases
\max\{y(n)-r,\inf_{\alpha\in\Lambda}x_\alpha(n)\}&\text{if }y(n)\geq r=
 \qquad\hfill\text{(1)}\\
\max\{0,\inf_{\alpha\in\Lambda}x_\alpha(n)\}&\text{if }0\leq=
 y(n)<r\qquad\hfill\text{(2)} \\
\min\{0,\sup_{\alpha\in\Lambda}x_\alpha(n)\}&\text{if }-r<y(n)<0=
 \qquad\hfill\text{(3)}\\
\min\{y(n)+r,\sup_{\alpha\in\Lambda}x_\alpha(n)\}&\text{if }y(n)\leq=
 -r\qquad\hfill\text{(4)} \endcases$$
If we perform this change (in the space $\ell_\infty$) from $y$ to $x$ one=
 coordinate at a time, it is not hard to see that our construction ensures=
 that $x\in A_\infty$. =20

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Since $\|x-y\|\leq r$, it follows  that=20
$$\|x-x_\alpha\|\leq r_\alpha+r\leq 2\, r_\alpha\text{ for all }=
 \alpha\in\Lambda .$$
Thus the proof of (4) will be complete once we show that $x\in c_0$. =20

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Suppose $x\not\in c_0$. Then there is an $\varepsilon>0$  and  a subsequence=
 $(n_k)\in\Bbb N$ such that $|x(n_k)|>\varepsilon$ for all $k\in\Bbb N$.
Let=20
$$N_1:=3D\{k\in\Bbb N :\ \text{Condition (1) in the definition of $x(n_k)$=
 applies}\}.$$ $N_2,N_3$ and $N_4$ are defined analogously.

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Suppose $k\in N_1$. Then=20
$$y(n_k)>r+\varepsilon\text{ or=
 }\inf_{\alpha\in\Lambda}x_\alpha(n_k)>\varepsilon.$$

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By our previous observation about $y$, and the fact that the centers=
 $x_\alpha$ lie in the space $c_0$, this can happen only finitely often.=
 Thus $N_1$ is finite.=20

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The proofs that $N_2,N_3$ and $N_4$ are also finite are similar. This yields=
 a contradiction to our assumption; consequently $x\in c_0$.
\qed\enddemo

\special{src: 430 KKNO626.TEX} %Inserted by TeXtelmExtel

\remark{Remark} It follows from Sobczyk's Theorem~\cite{Sob} that $E(c_0)=3D=
2$ and is exact. Since the proofs of Sobczyk's Theorem known to us do not=
 reveal whether the points of intersection still lie in the admissible set=
 containing the centers of the balls, we have presented our own=
 construction.=20
\endremark\medskip

\special{src: 435 KKNO626.TEX} %Inserted by TeXtelmExtel

The example  of $c_0$ above can be modified to obtain an example of  a=
 Banach space which is sequentially hyperconvex, but fails to be=
 hyperconvex. =20

\special{src: 439 KKNO626.TEX} %Inserted by TeXtelmExtel

\definition{Definition 3}
Let=20
$\ell_\infty^{c}([0,1])$ denote the closed subspace of $\ell_\infty([0,1])$,=
 defined by=20
$$x=3D(x_i)\in \ell_\infty^{c}([0,1]) \text{ if and only if } \{i\in [0,1]:\=
 x_i\not=3D0\} \text{ is countable.}$$
\enddefinition

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\proclaim{Theorem 5}=20
\roster
\runinitem $\Cal{A}(\ell_\infty^{c}([0,1]))$ fails the Intersection=
 Property.
\item $\ell_\infty^{c}([0,1])$ fails to be hyperconvex.=20
\item $\ell_\infty^{c}([0,1])$ is sequentially hyperconvex.
\item $\ell_\infty^{c}([0,1])$ is $2$-hyperconvex.
\endroster\endproclaim
=20
\special{src: 457 KKNO626.TEX} %Inserted by TeXtelmExtel
=20
=20
\special{src: 460 KKNO626.TEX} %Inserted by TeXtelmExtel
=20
\demo{Proof} (1) follows directly from (2) and (3). Let us first prove that
the space $\ell_\infty^{c}([0,1])$ is not hyperconvex.  Indeed, for any $j=
 \in [0,1]$, set
$e_j =3D (\delta_i^j)$ where $\delta_i^j =3D 0$ whenever $i \neq
j$ and $\delta_j^j =3D1$. It is easy to check that=20
$\|e_i - e_j \| =3D 1\;\;\text{for any $i \neq j$.}$=20
Whenever a  point $x =3D (x(i)) \in l_{\infty}([0,1])$ satisfies=20
$\|x - e_j\| \leq {1}/{2} \;\;\text{for all $j \in [0,1]$,}$
then $x$ will also satisfy=20
$x(i) \geq {1}/{2} \;\;\text{for all $i \in [0,1]$ .}$=20
Thus the  support of $x$ will not be countable. Hence
$$\ell_\infty^{c}([0,1]) \cap \Big(\bigcap_{j \in [0,1]} B(e_j,1/2)\Big) =3D=
 \emptyset,$$
which means $\ell_\infty^{c}([0,1])$ is not hyperconvex. =20

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Let us show next that $\ell_\infty^{c}([0,1])$ is
sequentially hyperconvex.  Indeed, let $(x_n)_{n \geq 1}$ be in=
 $\ell_\infty^{c}([0,1])$ such that
$$\|x_n - x_m\| \leq r_n + r_m \;\;\text{for any $n,m =3D 1,2,\ldots$}$$
for some positive numbers $(r_n)$.  Write $x_n =3D (x_n(i))_{i
\in [0,1]}$.  Set
$$I =3D \bigcup_{n \geq 1} \{i:\; x_n(i) \neq 0 \}\;.$$
By definition of the space $\ell_\infty^{c}([0,1])$, we deduce that $I$ is=
 countable.
Since $l_{\infty}(I)$ is hyperconvex, there exists $x=3D(x_i) \in
l_{\infty}(I)$ such that
$$\sup_{i \in I} \big|\big|x_i - x_n(i)\big|\big| \leq r_n$$
If we set $\hat{x} =3D (\hat{x}_i)$ defined by
$$\hat{x}_i =3D \cases
x_i & \text{if $i \in I$}\\
0 & \text{otherwise} \endcases $$
then we have
$$\hat{x} \in \ell_\infty^{c}([0,1]) \cap \Big(\bigcap_{n \geq 1}=
 B(x_n,r_n)\Big)$$
which completes the proof of our claim.

\special{src: 495 KKNO626.TEX} %Inserted by TeXtelmExtel

It remains to show that $\ell_\infty^{c}([0,1])$ is $2$-hyperconvex. Let $A$=
 be a nonempty admissible
subset of $\ell_\infty^{c}([0,1])$.  Let $x_{\alpha}\in A$, and $r_{\alpha}$=
 positive
numbers for $\alpha\in\Lambda$, such that
$$d(x_{\alpha},x_{\beta})\leq r_{\alpha}+r_{\beta} \text{ for every }\alpha,
\beta\in \Lambda\;.$$=20
Set $\displaystyle r =3D \inf_{\alpha \in \Lambda} r_{\alpha}$.  Hence there
exists a sequence $(r_{\alpha_n})$ such that
$\lim_{n \rightarrow +\infty} r_{\alpha_n} =3D r\;.$=20
Set=20
$$I =3D \bigcup_{n \geq 1} \{t \in [0,1]: x_{\alpha_n}(t) \neq 0\}$$
By the definition of the space $\ell_\infty^{c}([0,1])$, the set $I$ is=
 countable.
Clearly, we have
$x_{\alpha_n} \in l_{\infty}(I)\text{ for any $n \geq 1$}.$
Consider the family of admissible sets $\Cal{A}(l_{\infty}(I))$.   For any=
 $n \geq 1$, set $$A_n:=3D\operatorname{co}(x_{\alpha_m})_{m \geq n} =3D=
 \bigcap \{A \in \Cal{A}(l_{\infty}(I)):\ x_{\alpha_m} \in A\text{  for }=
 m\geq n\}\;.$$
Since $l_{\infty}(I)$ is hyperconvex, then the family =
 $\Cal{A}(l_{\infty}(I))$ satisfies the Intersection Property.  Therefore,=
 the set
$\displaystyle \bigcap_{n \geq 1} A_n$ is not empty.  Let=20
$$\omega \in \bigcap_{n \geq 1} A_n \subset l_{\infty}(I)\;.$$=20
Set
$$\omega_e =3D (\omega_e(t))_{t \in [0,1]} =3D
\cases
\omega(t) & \text{if $t \in I$}\\
0&\text{otherwise.}
\endcases $$
Clearly, we have $\omega_e \in \ell_\infty^{c}([0,1])$.  We claim that
$$\omega_e \in A \cap \Big(\bigcap_{\alpha \in \Lambda} B(x_{\alpha},=
 2r_{\alpha})\Big)$$
which will complete the proof of the that $\ell_\infty^{c}([0,1])$ is=
 $2$-hyperconvex.  First note that if $x \in
l_{\infty}(I)$ and $x_n \in B(x,r)$ for any $n \geq K$, for some
positive number $r$ and $K \geq 1$, then we must have $\omega \in
B(x,r)$.  This is the case because $\omega \in A_K \subset B(x,r)$.
On the other hand, if $x \in \ell_\infty^{c}([0,1])$ such that $x_{\alpha_n}=
 \in B(x,r)$ for any $n
\geq K$, for some $K \geq 1$.  Then we have
$$\omega_e \in B(x,r)$$
Indeed, let
$$x_r =3D (x_r(t))_{t \in I} =3D (x(t))_{t \in I} \in l_{\infty}(I)\;. $$
Hence the condition $x_n \in B(x,r)$ will imply
$$\align\Big|x_{\alpha_n}(t) - x(t) \Big| \leq r & \text{\quad for $t \in=
 I$}\\ &\\
\Big|x(t) \Big| \leq r & \text{\quad for $t \not\in I$}\\
\endalign$$
In particular, we have $x_{\alpha_n} \in B(x_r,r)$, for $n \geq K$, which
implies $\omega \in B(x_r,r)$ in view of the result above. Therefore,
we have
$$ \align
\Big|\omega(t) - x(t) \Big| \leq r & \text{\quad for $t \in I$}\\ &\\
\Big|x(t) \Big| \leq r & \text{\quad for $t \not\in I$}\\
\endalign$$
This clearly translates into $\omega_e \in B(x,r)$.  From this, we
conclude that
$$\omega_e \in  A\;.$$
Let us now show that $\omega_e \in B(x_{\alpha}, 2r_{\alpha})$ for any=
 $\alpha \in
\Lambda$.  Let $\varepsilon > 0$.  There exists $K \geq 1$ such that
$r_{\alpha_n} \leq r + \varepsilon$, for any $n \geq K$.  Hence, for
any $\alpha \in \Lambda$, we have
$$d\Big(x_{\alpha_n},x_{\alpha}\Big) \leq r_{\alpha_n}+r_{\alpha} \leq
r + \varepsilon + r_{\alpha}$$
for any $n \geq K$.  The above results imply=20
$$d\Big(\omega_e,x_{\alpha}\Big) \leq r + \varepsilon + r_{\alpha}$$
Since $\varepsilon$ was arbitrary, then we must have
$$d\Big(\omega_e,x_{\alpha}\Big) \leq r + r_{\alpha} \leq 2\,r_{\alpha}$$
for any $\alpha \in \Lambda$.  This  completes the proof that
$$\omega_e \in A \cap \Big(\bigcap_{\alpha \in \Lambda} B(x_{\alpha},=
 2r_{\alpha})\Big)$$
which in turn completes our proof.
\qed\enddemo
 =20
\special{src: 560 KKNO626.TEX} %Inserted by TeXtelmExtel
 =20

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=20
\special{src: 575 KKNO626.TEX} %Inserted by TeXtelmExtel
=20

\special{src: 578 KKNO626.TEX} %Inserted by TeXtelmExtel

\Refs
\widestnumber \key{JMP}

\special{src: 583 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref
\key Al
\by D.E. Alspach=20
\paper A Fixed Point Free Nonexpansive Map
\jour Proc. Amer. Math. Soc.
\vol 82 \yr 1981
\pages 423--424
\endref

\special{src: 594 KKNO626.TEX} %Inserted by TeXtelmExtel


\special{src: 597 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref=20
\key AP
\by N. Aronszajn, P. Panitchpakdi
 \paper Extensions of uniformly continuous transformations and hyperconvex=
 metric spaces
\jour Pacific J. Math. \vol  6 \yr 1956 \pages 405--439
\endref

\special{src: 606 KKNO626.TEX} %Inserted by TeXtelmExtel


\special{src: 609 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref=20
\key Ba
\by J.B. Baillon=20
\paper Nonexpansive mappings and hyperconvex spaces=20
\jour Contemp. Math. \vol 72 \yr 1988 \pages 11--19
\endref

\special{src: 618 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref=20
\key Gr1
\by B. Gr\"unbaum
\paper On some covering and intersection properties in Minkowski spaces
\jour  Pacific J. Math. \vol  9 \yr 1959 \pages 487-494
\endref

\special{src: 627 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref
\key Gr2
\bysame
\paper Some applications of expansion constants
\jour Pacific J. Math. \vol  10 \yr 1960 \pages 193--201
\endref

\special{src: 636 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref
\key Gr3
\bysame
\paper Projection constants
\jour Trans. Amer. Math. Soc. \vol95 \yr 1960 \pages 451--465
\endref

\special{src: 645 KKNO626.TEX} %Inserted by TeXtelmExtel


\special{src: 648 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref
\key Ju
\by H.W.E. Jung
\paper \"Uber die kleinste Kugel, die eine r\"aumliche Figur einschlie\ss t
\jour J. Reine Angew. Math. \vol 123 \yr 1901 \pages 241--257
\endref

\special{src: 657 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref
\key Kh
\by M.A. Khamsi
\paper On  Metric Spaces With Uniform  Normal Structure
\jour Proc. Amer. Math. Soc.
\vol 106 \yr 1989
\pages 723--726
\endref

\special{src: 668 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref
\key KLS
\by M.A. Khamsi, M. Lin, R.C. Sine \paper On the fixed points of
commuting nonexpansive maps in hyperconvex spaces
\jour J. Math. Anal. Appl. \vol 168 \yr 1992 \pages 372--380
\endref

\special{src: 677 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref=20
\key  KR
\by M.A. Khamsi, S. Reich=20
\paper Nonexpansive mappings and
semigroups in hyperconvex spaces \jour Math. Japon.\vol 35 \yr 1990 \pages=
 467--471
\endref

\special{src: 686 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref
\key Ki
\by W. Kirk \paperinfo Private communication
\endref

\special{src: 693 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref
\key Kl
\by V.L. Klee Jr.
\paper The support property of a convex set in a linear normed space
\jour Duke Math. J.
\vol 15 \yr 1948
\pages 767--772
\endref

\special{src: 704 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref
\key Lim
\by T.C. Lim=20
\paper Asymptotic Centers and Nonexpansive Mappings in Conjugate
Banach Spaces
\jour Pacific J. Math.
\vol 90\yr 1980
\pages 135--143
\endref

\special{src: 716 KKNO626.TEX} %Inserted by TeXtelmExtel


\special{src: 719 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref
\key LS
\by M. Lin, R.C. Sine
\paper Retractions on the fixed point set
of semigroups of nonexpansive maps in hyperconvex spaces \toappear
\endref

\special{src: 728 KKNO626.TEX} %Inserted by TeXtelmExtel


\special{src: 731 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref
\key Pe
\by J.P. Penot=20
\paper Fixed point theorems without convexity
\jour Bull. Soc. Math. France
\vol 60
\yr 1979
\pages 129--152
\endref

\special{src: 743 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref
\key Si1
\by  R.C. Sine
\paper Hyperconvexity and approximate fixed points
\jour Nonlinear Anal. \vol 13 \yr 1989\pages 863--869
\endref

\special{src: 752 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref=20
\key Si2
\bysame \paper On linear contraction semigroups in sup
norm spaces \jour Nonlinear Anal. \vol 3 \yr 1979 \pages 885--890
\endref

\special{src: 760 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref=20
\key Soa
\by P. Soardi=20
\paper Existence of fixed points of nonexpansive
mappings in certain Banach lattices \jour Proc. Amer. Math. Soc. \vol73
\yr 1979  \pages 25--29
\endref

\special{src: 770 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref
\key Sob
\by A. Sobczyk=20
\paper Projection of the space $(m)$ on its subspace $(c_0)$=20
\jour Bull. Amer. Math. Soc.=20
\vol 47 \yr 1941 \pages  938--947
\endref
\endRefs

\special{src: 781 KKNO626.TEX} %Inserted by TeXtelmExtel

\enddocument

\special{src: 785 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref =20
\key Is=20
\by J.R. Isbell=20
\paper Six theorems about injective metric spaces=20
\jour Comment. Math. Helvetici \vol 39 \yr 1964 \pages 439--447
\endref

\special{src: 794 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref \key La=20
\by H.E. Lacey \book The isometric theory of classical Banach
spaces  \publ Springer-Berlin, Heidelberg-New York \yr 1974
\endref

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\special{src: 804 KKNO626.TEX} %Inserted by TeXtelmExtel

\ref
\key Lin
\by J. Lindenstrauss
\paper On nonlinear projections in Banach spaces
\jour Mich. Math. J.
\vol 11\yr 1964\pages 263--287
\endref

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\special{src: 817 KKNO626.TEX} %Inserted by TeXtelmExtel


\special{src: 820 KKNO626.TEX} %Inserted by TeXtelmExtel

Theorem~1 establishes that spaces which satisfy Penot's Intersection=
 Property are $2$-hyperconvex. (Thus reflexive Banach spaces and dual Banach=
 spaces are $2$-hyperconvex.) On the other hand, in Section~5, we construct=
 two examples of  $2$-hyperconvex spaces, which fail the Intersection=
 Property.  =20

\special{src: 824 KKNO626.TEX} %Inserted by TeXtelmExtel


Helmut Knaust
The University of Texas at El Paso
Department of Mathematical Sciences
El Paso TX 79968-0514             (915) 747-7002
<http://www.math.utep.edu/Faculty/helmut>


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 amstex
\documentstyle{amsppt}
\magnification=3D\magstep1
\pagewidth{6.5truein}
\pageheight{9.0truein}
\NoBlackBoxes
%%%%%%%%%%%
\topmatter
\title=20
Lambda-Hyperconvexity in Metric Spaces
\endtitle
\rightheadtext{LAMBDA-HYPERCONVEXITY IN METRIC SPACES}
\date  {\it Preliminary Version\qquad June 26, 1998}\enddate
\author M.A. Khamsi, H. Knaust, N.T. Nguyen and M.D. O'Neill \endauthor
\address Department of Mathematical Sciences, The University of Texas at El=
 Paso, El Paso, TX~79968-0514, USA\endaddress
\email mohamed\@math.utep.edu\endemail
\address Department of=20Mathematical Sciences, The University of Texas at=
 El Paso, El Paso, TX~79968-0514, USA\endaddress
\email helmut\@math.utep.edu\endemail
\address Department of Mathematical Sciences, The University of Texas at El=
 Paso, El Paso, TX~79968-0514, USA\endaddress
\email nguyen\@math.utep.edu\endemail
\address Department of Mathematical Sciences, The University of Texas at El=
 Paso, El Paso, TX~79968-0514, USA\endaddress
\email michael\@math.utep.edu\endemail

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\abstract=20
We introduce the concept of $\lambda$-hyperconvexity in metric spaces,=
 generalizing the classical notion of a hyperconvex metric space.  We show=
 that a bounded metric space which is $\lambda$-hyperconvex has the fixed=
 point property for nonexpansive mappings provided $\lambda < 2$.  Uniformly=
 convex Banach spaces are examples of such $\lambda$-hyperconvex spaces for=
 some $\lambda < 2$. We furthermore investigate the relationship between=
 Penot's Intersection Property and $2$-hyperconvexity.
\endabstract
\keywords
Hyperconvexity, fixed point
\endkeywords
\subjclass  Primary
46B20, 47E10\endsubjclass
\endtopmatter

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\heading{\S 1. Introduction}\endheading
Aronszajn and Panitchpakdi \cite{AP} define a metric space to be {\it=
 hyperconvex} if it satisfies the following condition:  Whenever a=
 collection of balls intersects pairwise, the intersection of all balls in=
 the collection is not empty. We generalize this concept by calling  a=
 metric space {\it $\lambda$-hyperconvex} if {\it every pairwise=
 intersecting collection of balls with centers in a given admissible set has=
 non-empty intersection in the admissible set if the radii of the balls are=
 increased by  the factor $\lambda$} (see Definition~2 below for a precise=
 definition). While the Hilbert space $\ell_2$ fails to be hyperconvex, it=
 naturally satisfies the condition of $\lambda$-hyperconvexity for =
 $\lambda=3D\sqrt{2}$.=20

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The
recent interest into hyperconvex spaces goes back to  results of Sine
\cite{Si2} and Soardi \cite{Soa} who proved independently that the fixed=
 point property for nonexpansive mappings holds in bounded hyperconvex=
 spaces.  Since then many interesting results \cite{Ba, KLS, KR, LS, Si1}=
 have been
shown to hold in hyperconvex spaces. We show in Section~2 that for=
 $\lambda<2$ the fixed point property holds also for nonexpansive mappings=
 in bounded $\lambda$-hyperconvex  spaces.=20

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In Section~3 we show that uniformly convex Banach  spaces are=
 $\lambda$-hyperconvex for some $\lambda<2$, while in Section~4 we=
 investigate the relationship between Penot's Intersection Property and=
 $2$-hyperconvexity.

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\definition{Definition~1} A metric space $(M,d)$ is {\it hyperconvex} if and=
 only if for
any family of points $(x_\alpha)_{\alpha \in \Lambda}$ and any family of=
 positive
numbers $(r_\alpha)_{\alpha \in \Lambda}$ such that
$d(x_\alpha,x_\beta) \leq r_\alpha + r_\beta \text{ for all } \alpha,\beta=
 \in \Lambda$
we must have
$$\bigcap_{\alpha \in \Lambda} B(x_\alpha,r_\alpha) \neq \emptyset, $$
where $B(x,r)$ denotes the closed ball centered at $x \in M$ with
radius $r$.=20
\enddefinition
$\ell_\infty(I)$ for any index set $I$ is the classical example of a=
 hyperconvex metric space, while the Hilbert space $\ell_2$ fails to be=
 hyperconvex.

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We may refine this definition by considering the cardinality of the
index set $\Lambda$.  We will say that $(M,d)$ (or $M$ for
short) is {\it finitely hyperconvex} (resp. {\it sequentially hyperconvex})=
 if the
property above is satisfied for any finite set $\Lambda$ (resp. countable
set $\Lambda$). =20

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Recall that a subset $A$ of $M$ is called {\it admissible} if and only
if $A$ is an intersection of closed balls.  The family of admissible
subsets of $M$ will be  denoted by $\Cal{A}(M)$.

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This article is centered around the following notion:
\definition{Definition 2} Let $M$ be a metric space and
let $\lambda\geq 1$.  We say that the metric space $M$ is {\it
$\lambda$-hyperconvex} if for every non-empty admissible set $A$ of
$M$, for any family of closed balls
$\{B(x_{\alpha},r_{\alpha})\}_{\alpha\in\Lambda}$, each of radius=20
$r_{\alpha}$, centered at $x_{\alpha}\in A$ for $\alpha\in\Lambda$, the
condition
$d(x_{\alpha},x_{\beta})\leq r_{\alpha}+r_{\beta} \text{ for every }\alpha,
\beta\in \Lambda,$=20
implies=20
$$A\cap \Big(\bigcap_{\alpha\in\Lambda} B(x_{\alpha},\lambda=
 r_{\alpha})\Big)\neq
\emptyset.$$
We let $\Lambda(M)$ be the infimum of all constants $\lambda$ such that $M$=
 is $\lambda$-hyperconvex, and say that $\Lambda(M)$ is {\it exact} if $M$=
 is $\Lambda(M)$-hyperconvex.
\enddefinition

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Gr\"unbaum~\cite{Gr1,Gr2,Gr3} and other authors have studied a similar=
 property not involving the
underlying admissible set $A$.  For our purposes it is essential to include=
 the restriction that the intersection of the expanded balls also intersects=
 with the admissible set containing the centers of the original balls.=
 Otherwise we will not be able to connect this concept to the normal=
 structure property (see the proof
of Theorem~2 for more details).

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Let us recall Gr\"unbaum's definition: For a metric space $M$, let the {\it=
 expansion constant} $E(M)$ be the infimum of all constants $\mu$ such that=
 the following holds: Whenever a collection $\{B(x_\alpha,r_\alpha) :\=
 \alpha\in \Lambda\}$ intersects pairwise, then
$$ \bigcap_{\alpha\in\Lambda}B(x_\alpha,\mu\cdot r_\alpha)\not=3D\emptyset.$=
$
We say $E(M)$ is {\it exact}, if the condition is even satisfied for=
 $\mu=3DE(M)$. =20

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Trivially, $E(M)\leq\Lambda(M)$ holds in metrically convex spaces. On the=
 other hand, if $M$ is a two element metric space, then $E(M)=3D1$, while=
 $\Lambda(M)=3D2$, so both concepts do not coincide in general.

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Let us first summarize some basic properties of $\lambda$-hyperconvex metric=
 spaces, some of which are trivial, while the others can be easily derived=
 from corresponding results about expansion constants:=20

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\proclaim{Theorem~1} \roster
\runinitem A metric space is hyperconvex if and only if it is=
 $1$-hyperconvex.
\item For every complete metric space $M$, $\Lambda(M)\leq 2$~\cite{Gr2}.=
 Every $\lambda$-hyperconvex metric space is complete. =20
\item   If $\Cal{A}(M)$ has the Intersection Property, then $\Lambda(M)$ is=
 exact~\cite{Gr2, Theorem~3}.
Here we say, $\Cal{A}(M)$ has the {\it Intersection Property} if and only if=
 any family of elements $\{A_{\alpha}\}_{\alpha
\in \Lambda}$ in $\Cal{A}(M)$ has a nonempty intersection provided=20
$$\bigcap_{\alpha\in \Lambda_f} A_\alpha\neq \emptyset$$
for any finite subset $\Lambda_f$ of $\Lambda$~\cite{Pe}.
\item In particular, reflexive Banach spaces and dual Banach spaces are=
 $2$-hyperconvex.=20
\item There is a subspace $X$ of $\ell_1$, which fails to be=
 $2$-hyperconvex~\cite{Gr2}.
 \item Hilbert space is $\sqrt{2}$-hyperconvex~\cite{Ju, Gr1}.
\endroster\endproclaim
\remark{Remark}The counterexample in (5) is the following example, first=
 considered by Klee~\cite{Kl} in a different context:
$$X=3D\left\{(x_n)_{n\in\Bbb N} \in \ell_1:\ \sum_{n=3D1}^\infty=
 \frac{n}{n+1} x_n=3D0\right\}.$$
\endremark

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\heading{\S 2. $\lambda$-Hyperconvexity and the Fixed Point=
 Property}\endheading
We establish the following generalization of the Fixed Point Theorem by=
 Sine~\cite{Si2} and Soardi~\cite{Soa}:

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\proclaim{Theorem 2}  Let $M$ be a bounded $\lambda$-hyperconvex space. If
$\lambda<2$, then
any nonexpansive map $f:M\rightarrow M$ has a fixed point.
\endproclaim
A map $f:M\rightarrow M$ is {\it  nonexpansive} if $$
d(f(x),f(y))\leq d(x,y) \text{ for every } x, y \in M, $$
and  if=20
$$
d(f(x),f(y))< d(x,y) \text{ for every } x, y \in M \text{ with } x\neq y,
$$
then   $f$ is called  {\it=20
strictly nonexpansive}.

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The proof of Theorem 2 makes use of the following notation:  Let $A$ be  a=
 bounded subset of a metric  space $M$. We set=20
$$\align=20
R(x,A)&:=3D\sup\{d(x,y): y\in A\} \text{ for } x\in M, \\
\delta(A)&:=3D\sup\{R(x,A): x\in A\}  \\
\text{and}\quad R(A)&:=3D\inf\{R(x,A): x\in A\}
\endalign$$
We also need the following definitions:
Let $\Cal F$ be a family of subsets in a
metric  space $M$. We say that $\Cal F$ defines a  {\it convexity structure=
 on $M$}
if it contains the balls and  is stable  by  intersection.

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Assume that $M$ is  a
metric  space with a convexity structure $\Cal F$. We say that
$\Cal F$ is  a {\it uniform
normal structure on $M$}
if there exists $c<1$ such that=20
$$R(A)\leq c\,\delta(A) \text{ for every } A\in \Cal F.$$=20

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The proof of Theorem 2 is based on the following fact established=
 in~\cite{Kh}.
\proclaim{Theorem} Let $M$ be a bounded complete metric space.
If $M$ has a uniform normal structure, then $M$ has the fixed point property
for nonexpansive maps.
\endproclaim

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\demo{Proof of Theorem 2} Let $M$ be a bounded
$\lambda$-hyperconvex space, with  $\lambda<2$. By a previous remark, $M$ is=
 complete. By Khamsi's result it
suffices to prove that $M$ has a uniform normal structure.=20

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 The family of admissible sets $\Cal{A}(M)$
defines a convexity structure on $M$.

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We shall   show that $\Cal{A}(M)$ is a uniform  normal structure on
$M$. Let $A\in \Cal A(M)$ with $\delta(A)>0$.
For every $x\in A$, let $B(x,r_x)$ denote the ball
centered at $x$ with constant radius $r_x=3D\frac{1}{2}\delta(A)$. Then we=
 have=20
$$d(x,y)\leq \delta(A)=3D
\frac{1}{2}\delta(A)+\frac{1}{2}\delta(A)=3Dr_x+r_y\text{ for every } x,
y \in A.$$
Since $M$ is  $\lambda$-hyperconvex  we can find =20
$$x_0\in A\cap\bigcap_{\alpha\in\Lambda} B(x_{\alpha},\lambda
r_{\alpha}).$$
Thus we have
$$d(x,x_0)\leq\frac{1}{2}\lambda\,\delta(A)\text{ for every } x \in A.
$$
It follows that=20
$$R(A)\leq\frac{1}{2}\lambda\,\delta(A).$$
Since $\frac{1}{2}\lambda<1$, $\Cal{A}(M)$ is a uniform normal structure on=
 $M$ and the
theorem is proved.
\qed\enddemo

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\remark{Remark}Alspach~\cite{Al} (see also   Lim~\cite{Lim}) constructed an=
=20
example of a weakly compact convex set in $L^1[0,1]$ which fails the fixed
point property for an isometry. From Theorem~1 we can conclude
that this example is $2$-hyperconvex,  while Theorem~2 shows that this=
 example is not $\lambda$-hyperconvex for any $\lambda<2$. In particular,=
 Theorem~2 fails for $\lambda=3D2$.\medskip
\endremark

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\heading{\S 3. Uniformly Convex Spaces}\endheading
In this section, we prove the following result:
\proclaim{Theorem 3} A uniformly convex Banach space
is $\lambda$-hyperconvex for some $\lambda < 2$.
\endproclaim

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\demo{Proof} Let $X$ be a uniformly convex Banach
space.  Since $X$ is reflexive, it is enough to show that for any
admissible subset $A$, any choice of points $x_1, \ldots, x_n \in A$ and
positive numbers $r_1, \ldots, r_n$ such that
$$d(x_i,x_j) \leq r_i + r_j\;\;\text{for all $i,j =3D 1, \ldots, n$},$$
we have
$$A \cap \Big(\bigcap_{1 \leq i \leq n} B(x_i,\lambda
r_i)\Big) \neq \emptyset$$
for some $\lambda < 2$ independent of the set $A$ and the points
$\{x_i\}$ as well as the radii $\{r_i\}$.  Without loss of
generality, we may assume that $r_1 \leq r_2 \leq \ldots \leq r_n$.
We know from the proof of Part (3) of Theorem~1 that=20
$$x_1 \in \bigcap_{1 \leq i \leq n} B(x_i,2\,r_i)\;.$$
We may assume that there exists $i\in\{1,\ldots,n\}$ such that $d(x_1, x_i)
> r_1$.  Otherwise, we have
$$x_1 \in \bigcap_{1 \leq i \leq n} B(x_i,r_1)  \subset \bigcap_{1
\leq i \leq n} B(x_i,r_i) \subset \bigcap_{1 \leq i \leq n}
B(x_i,\lambda r_i)$$
for any $\lambda \geq 1$.  Set $\Lambda =3D \{i:\; d(x_1, x_i) > r_1\}$.
Let $i_0$ be the smallest element of $\Lambda$.  Then we know that=20
$$x_{i_0} \in \bigcap_{j \in \Lambda} B(x_j,2\,r_j)$$
Since $d(x_1, x_{i_0}) > r_1$, then there exists $x_* \in
\text{Segment}[x_1, x_{i_0}]$ such that $d(x_1, x_*) =3D r_1$.  For any
$i \not\in \Lambda$, then we have=20
$$d(x_*,x_i) \leq d(x_*,x_1) + d(x_1,x_i) \leq 2r_1 \leq 2r_i \;.$$
This clearly implies
$$x_* \in \Big(\bigcap_{1 \leq i \leq n} B(x_i,2\,r_i)\Big) \cap=
 \operatorname{conv}(x_i)$$
where $\operatorname{conv}(x_i)$ is the convex hull of the points
$x_1,\ldots,x_n$.  Let $\varepsilon > 0$.  Let $i\in\{1,\ldots,n\}$.  We=
 consider
two cases:

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\noindent $\underline{\text{\bf Case 1:}}\  r_1 < \varepsilon\; r_i\;.$

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\noindent
Then we have
$$d\left(x_i, \frac{x_1+x_*}{2} \right) \leq d(x_i,x_1) + d\left(x_1,
\frac{x_1+x_*}{2} \right) \leq r_i + r_1 + \frac{r_1}{2} =3D \frac{2r_i
+ 3r_1}{2}$$
Using the above condition on $r_1$ and $r_i$, we get
$$d\left(x_i, \frac{x_1+x_*}{2} \right) \leq \frac{2+=
 3\varepsilon}{2}\;r_i$$

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\noindent $\underline{\text{\bf Case 2:}}\   \varepsilon\; r_i \leq r_1\;.$

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\noindent Since
$$d(x_1,x_*) =3D r_1 \geq 2\,r_i \frac{\varepsilon}{2}$$
and
$$d(x_i,x_1) \leq 2r_i \;\;\text{as well as}\;\;d(x_i,x_*) \leq 2r_i\;,$$=20
we get from the definition of the modulus of uniform convexity
$$d\left(x_i, \frac{x_1+x_*}{2} \right) \leq 2\,r_i \Big(1 - \delta
\left({\varepsilon}/{2}\right)\Big)\;.$$ \smallskip

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\noindent Set
$$\lambda =3D 2\, \min_{\varepsilon > 0}\left\{\max\left\{\frac{2+=
 3\varepsilon}{4},1 - \delta
\left({\varepsilon}/{2}\right)\right\}\right\}$$
Then we have
$$\frac{x_1+x_*}{2} \in=20
\Big(\bigcap_{1 \leq i \leq n} B(x_i,\lambda\, r_i)\Big) \cap
\operatorname{conv}(x_i) \subset A \cap \Big(\bigcap_{1 \leq i \leq n}=
 B(x_i,\lambda\,
r_i)\Big),$$
because $\operatorname{conv}(x_i) \subset A$.  Let us show that $\lambda <=
 2$.
Indeed, for $\varepsilon < 2/3$, we have=20
$$\max\left\{\frac{2+ 3\varepsilon}{4},1 - \delta
\left({\varepsilon}/{2}\right)\right\} < 1$$
which clearly implies $\lambda < 2$.  The proof of our claim is
therefore complete.
\qed\enddemo

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\remark{Remarks} =20
A look at the proof reveals that one  does not need the assumption that $X$=
 is uniformly convex.  It is enough to assume that the {\it characteristic=
 of uniform convexity of
$X$} is less than 1/3, that is
$$\varepsilon_0(X):=3D \sup \{\varepsilon \ :\  \delta(\varepsilon) =3D=
 0\}\; < \frac{1}{3} \;.$$
We do not know whether the above result still holds if we only assume that=
=20
$\varepsilon_0(X) < 1$.
\endremark

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\heading{\S 4. Hyperconvexity and the Intersection Property}\endheading
Clearly if $M$ is  a hyperconvex space, then $\Cal{A}(M)$ has the=
 Intersection Property. Kirk~\cite{Ki} asks whether this is still true in=
 the case of $2$-hyperconvexity. We will show that $c_0$ provides a=
 counterexample to this question.

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\proclaim{Theorem 4}=20
\roster
\runinitem $c_0$ fails the Intersection Property.
\item $c_0$ fails to be hyperconvex.=20
\item $c_0$ is finitely hyperconvex.
\item $c_0$ is $2$-hyperconvex.
\endroster\endproclaim

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\demo{Proof} Note that (1) is a direct consequence of  (2) and (3).
It is easy to find an example which shows that $c_0$ fails to be=
 hyperconvex, establishing  (2).

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To prove (3), let
$x_1,\ldots,x_K$ be elements in $c_0$ and $r_1,\ldots r_K$ be positive=
 numbers such
that
$$\|x_i - x_j \| \leq r_i + r_j$$
for $i,j =3D 1,2, \ldots, K$.  We may assume that $0 < r_1 \leq r_2 \leq
\cdots \leq r_K$.  If we write $x_i =3D (x_i(n))$, then there exists $n_0
\geq 1$ such that $|x_i(n)| \leq r_1$ for all $i=3D1,\ldots,K$, and all $n=
 \geq
n_0$.  Let $x \in c_0$ defined by=20
$$x(n) \in \Big[\max_{1 \leq i \leq n_0} (x_i - r_i)\;,\;\min_{1 \leq
i \leq n_0} (x_i + r_i)\Big]\text{ for $1\leq n\leq n_0$,}$$ and=
 $$x(n)=3D0\text{ for $n >n_0$.}$$
Clearly, each of the intervals used in the definition is non-empty, and
$$x \in \bigcap_{1 \leq i \leq K} B(x_i,r_i)\;,$$ establishing (3).

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It remains to prove that $c_0$ is $2$-hyperconvex.
Let $A$ be an admissible set in $c_0$, and let $A_\infty$ denote the=
 intersection of the same balls in $\ell_\infty$, {\it i.e.,}=
 $A=3DA_\infty\cap c_0$.
Let a collection of balls $\{B(x_\alpha,r_\alpha)\}_{\alpha\in\Lambda}$ be=
 given, each centered at some $x_\alpha\in A$ with positive radius=
 $r_\alpha$. We assume that the collection satisfies the conditions
$$\|x_\alpha-x_\beta\|\leq r_\alpha + r_\beta\text{ for all=
 }\alpha,\beta\in\Lambda.$$

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\noindent  Since $\ell_\infty$ is hyperconvex, we can find $y=3D(y(n))\in=
 A_\infty$
such that=20
$$\|y-x_\alpha\|\leq r_\alpha \text{ for all } \alpha\in\Lambda.$$=20
If we write $a=3D(a(n))$ for an element in $A_\infty$, then=20
$$\{a(n)\ :\ a\in A_\infty\}$$ is a closed interval on the real line for all=
 $n\in\Bbb N$. Thus we may assume that=20
$$\inf_{\alpha\in\Lambda}x_\alpha(n)\leq y(n)\leq=
 \sup_{\alpha\in\Lambda}x_\alpha(n)\text{ for all } n\in\Bbb N.$$
\noindent Set $r=3D\inf_{\alpha\in\Lambda} r_\alpha$.  Note that for every=
 $\varepsilon>0$, there is an $N\in\Bbb N$ such that for all $n\geq N$
$$|y(n)|\leq r+\varepsilon.$$
Indeed, pick ${\alpha_0}\in\Lambda$ such that $r_{\alpha_0}<r+\varepsilon$ =
 and observe that $$\lim_{n\to\infty}x_{\alpha_0}(n)=3D0.$$

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Next we define $x=3D(x(n))\in\ell_\infty$ as follows:
$$x(n):=3D\cases
\max\{y(n)-r,\inf_{\alpha\in\Lambda}x_\alpha(n)\}&\text{if }y(n)\geq r=
 \qquad\hfill\text{(1)}\\
\max\{0,\inf_{\alpha\in\Lambda}x_\alpha(n)\}&\text{if }0\leq=
 y(n)<r\qquad\hfill\text{(2)} \\
\min\{0,\sup_{\alpha\in\Lambda}x_\alpha(n)\}&\text{if }-r<y(n)<0=
 \qquad\hfill\text{(3)}\\
\min\{y(n)+r,\sup_{\alpha\in\Lambda}x_\alpha(n)\}&\text{if }y(n)\leq=
 -r\qquad\hfill\text{(4)} \endcases$$
If we perform this change (in the space $\ell_\infty$) from $y$ to $x$ one=
 coordinate at a time, it is not hard to see that our construction ensures=
 that $x\in A_\infty$. =20

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Since $\|x-y\|\leq r$, it follows  that=20
$$\|x-x_\alpha\|\leq r_\alpha+r\leq 2\, r_\alpha\text{ for all }=
 \alpha\in\Lambda .$$
Thus the proof of (4) will be complete once we show that $x\in c_0$. =20

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Suppose $x\not\in c_0$. Then there is an $\varepsilon>0$  and  a subsequence=
 $(n_k)\in\Bbb N$ such that $|x(n_k)|>\varepsilon$ for all $k\in\Bbb N$.
Let=20
$$N_1:=3D\{k\in\Bbb N :\ \text{Condition (1) in the definition of $x(n_k)$=
 applies}\}.$$ $N_2,N_3$ and $N_4$ are defined analogously.

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Suppose $k\in N_1$. Then=20
$$y(n_k)>r+\varepsilon\text{ or=
 }\inf_{\alpha\in\Lambda}x_\alpha(n_k)>\varepsilon.$$

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By our previous observation about $y$, and the fact that the centers=
 $x_\alpha$ lie in the space $c_0$, this can happen only finitely often.=
 Thus $N_1$ is finite.=20

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The proofs that $N_2,N_3$ and $N_4$ are also finite are similar. This yields=
 a contradiction to our assumption; consequently $x\in c_0$.
\qed\enddemo

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\remark{Remark} It follows from Sobczyk's Theorem~\cite{Sob} that $E(c_0)=3D=
2$ and is exact. Since the proofs of Sobczyk's Theorem known to us do not=
 reveal whether the points of intersection still lie in the admissible set=
 containing the centers of the balls, we have presented our own=
 construction.=20
\endremark\medskip

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The example  of $c_0$ above can be modified to obtain an example of  a=
 Banach space which is sequentially hyperconvex, but fails to be=
 hyperconvex. =20

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\definition{Definition 3}
Let=20
$\ell_\infty^{c}([0,1])$ denote the closed subspace of $\ell_\infty([0,1])$,=
 defined by=20
$$x=3D(x_i)\in \ell_\infty^{c}([0,1]) \text{ if and only if } \{i\in [0,1]:\=
 x_i\not=3D0\} \text{ is countable.}$$
\enddefinition

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\proclaim{Theorem 5}=20
\roster
\runinitem $\Cal{A}(\ell_\infty^{c}([0,1]))$ fails the Intersection=
 Property.
\item $\ell_\infty^{c}([0,1])$ fails to be hyperconvex.=20
\item $\ell_\infty^{c}([0,1])$ is sequentially hyperconvex.
\item $\ell_\infty^{c}([0,1])$ is $2$-hyperconvex.
\endroster\endproclaim
=20
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=20
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=20
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=20
\demo{Proof} (1) follows directly from (2) and (3). Let us first prove that
the space $\ell_\infty^{c}([0,1])$ is not hyperconvex.  Indeed, for any $j=
 \in [0,1]$, set
$e_j =3D (\delta_i^j)$ where $\delta_i^j =3D 0$ whenever $i \neq
j$ and $\delta_j^j =3D1$. It is easy to check that=20
$\|e_i - e_j \| =3D 1\;\;\text{for any $i \neq j$.}$=20
Whenever a  point $x =3D (x(i)) \in l_{\infty}([0,1])$ satisfies=20
$\|x - e_j\| \leq {1}/{2} \;\;\text{for all $j \in [0,1]$,}$
then $x$ will also satisfy=20
$x(i) \geq {1}/{2} \;\;\text{for all $i \in [0,1]$ .}$=20
Thus the  support of $x$ will not be countable. Hence
$$\ell_\infty^{c}([0,1]) \cap \Big(\bigcap_{j \in [0,1]} B(e_j,1/2)\Big) =3D=
 \emptyset,$$
which means $\ell_\infty^{c}([0,1])$ is not hyperconvex. =20

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Let us show next that $\ell_\infty^{c}([0,1])$ is
sequentially hyperconvex.  Indeed, let $(x_n)_{n \geq 1}$ be in=
 $\ell_\infty^{c}([0,1])$ such that
$$\|x_n - x_m\| \leq r_n + r_m \;\;\text{for any $n,m =3D 1,2,\ldots$}$$
for some positive numbers $(r_n)$.  Write $x_n =3D (x_n(i))_{i
\in [0,1]}$.  Set
$$I =3D \bigcup_{n \geq 1} \{i:\; x_n(i) \neq 0 \}\;.$$
By definition of the space $\ell_\infty^{c}([0,1])$, we deduce that $I$ is=
 countable.
Since $l_{\infty}(I)$ is hyperconvex, there exists $x=3D(x_i) \in
l_{\infty}(I)$ such that
$$\sup_{i \in I} \big|\big|x_i - x_n(i)\big|\big| \leq r_n$$
If we set $\hat{x} =3D (\hat{x}_i)$ defined by
$$\hat{x}_i =3D \cases
x_i & \text{if $i \in I$}\\
0 & \text{otherwise} \endcases $$
then we have
$$\hat{x} \in \ell_\infty^{c}([0,1]) \cap \Big(\bigcap_{n \geq 1}=
 B(x_n,r_n)\Big)$$
which completes the proof of our claim.

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It remains to show that $\ell_\infty^{c}([0,1])$ is $2$-hyperconvex. Let $A$=
 be a nonempty admissible
subset of $\ell_\infty^{c}([0,1])$.  Let $x_{\alpha}\in A$, and $r_{\alpha}$=
 positive
numbers for $\alpha\in\Lambda$, such that
$$d(x_{\alpha},x_{\beta})\leq r_{\alpha}+r_{\beta} \text{ for every }\alpha,
\beta\in \Lambda\;.$$=20
Set $\displaystyle r =3D \inf_{\alpha \in \Lambda} r_{\alpha}$.  Hence there
exists a sequence $(r_{\alpha_n})$ such that
$\lim_{n \rightarrow +\infty} r_{\alpha_n} =3D r\;.$=20
Set=20
$$I =3D \bigcup_{n \geq 1} \{t \in [0,1]: x_{\alpha_n}(t) \neq 0\}$$
By the definition of the space $\ell_\infty^{c}([0,1])$, the set $I$ is=
 countable.
Clearly, we have
$x_{\alpha_n} \in l_{\infty}(I)\text{ for any $n \geq 1$}.$
Consider the family of admissible sets $\Cal{A}(l_{\infty}(I))$.   For any=
 $n \geq 1$, set $$A_n:=3D\operatorname{co}(x_{\alpha_m})_{m \geq n} =3D=
 \bigcap \{A \in \Cal{A}(l_{\infty}(I)):\ x_{\alpha_m} \in A\text{  for }=
 m\geq n\}\;.$$
Since $l_{\infty}(I)$ is hyperconvex, then the family =
 $\Cal{A}(l_{\infty}(I))$ satisfies the Intersection Property.  Therefore,=
 the set
$\displaystyle \bigcap_{n \geq 1} A_n$ is not empty.  Let=20
$$\omega \in \bigcap_{n \geq 1} A_n \subset l_{\infty}(I)\;.$$=20
Set
$$\omega_e =3D (\omega_e(t))_{t \in [0,1]} =3D
\cases
\omega(t) & \text{if $t \in I$}\\
0&\text{otherwise.}
\endcases $$
Clearly, we have $\omega_e \in \ell_\infty^{c}([0,1])$.  We claim that
$$\omega_e \in A \cap \Big(\bigcap_{\alpha \in \Lambda} B(x_{\alpha},=
 2r_{\alpha})\Big)$$
which will complete the proof of the that $\ell_\infty^{c}([0,1])$ is=
 $2$-hyperconvex.  First note that if $x \in
l_{\infty}(I)$ and $x_n \in B(x,r)$ for any $n \geq K$, for some
positive number $r$ and $K \geq 1$, then we must have $\omega \in
B(x,r)$.  This is the case because $\omega \in A_K \subset B(x,r)$.
On the other hand, if $x \in \ell_\infty^{c}([0,1])$ such that $x_{\alpha_n}=
 \in B(x,r)$ for any $n
\geq K$, for some $K \geq 1$.  Then we have
$$\omega_e \in B(x,r)$$
Indeed, let
$$x_r =3D (x_r(t))_{t \in I} =3D (x(t))_{t \in I} \in l_{\infty}(I)\;. $$
Hence the condition $x_n \in B(x,r)$ will imply
$$\align\Big|x_{\alpha_n}(t) - x(t) \Big| \leq r & \text{\quad for $t \in=
 I$}\\ &\\
\Big|x(t) \Big| \leq r & \text{\quad for $t \not\in I$}\\
\endalign$$
In particular, we have $x_{\alpha_n} \in B(x_r,r)$, for $n \geq K$, which
implies $\omega \in B(x_r,r)$ in view of the result above. Therefore,
we have
$$ \align
\Big|\omega(t) - x(t) \Big| \leq r & \text{\quad for $t \in I$}\\ &\\
\Big|x(t) \Big| \leq r & \text{\quad for $t \not\in I$}\\
\endalign$$
This clearly translates into $\omega_e \in B(x,r)$.  From this, we
conclude that
$$\omega_e \in  A\;.$$
Let us now show that $\omega_e \in B(x_{\alpha}, 2r_{\alpha})$ for any=
 $\alpha \in
\Lambda$.  Let $\varepsilon > 0$.  There exists $K \geq 1$ such that
$r_{\alpha_n} \leq r + \varepsilon$, for any $n \geq K$.  Hence, for
any $\alpha \in \Lambda$, we have
$$d\Big(x_{\alpha_n},x_{\alpha}\Big) \leq r_{\alpha_n}+r_{\alpha} \leq
r + \varepsilon + r_{\alpha}$$
for any $n \geq K$.  The above results imply=20
$$d\Big(\omega_e,x_{\alpha}\Big) \leq r + \varepsilon + r_{\alpha}$$
Since $\varepsilon$ was arbitrary, then we must have
$$d\Big(\omega_e,x_{\alpha}\Big) \leq r + r_{\alpha} \leq 2\,r_{\alpha}$$
for any $\alpha \in \Lambda$.  This  completes the proof that
$$\omega_e \in A \cap \Big(\bigcap_{\alpha \in \Lambda} B(x_{\alpha},=
 2r_{\alpha})\Big)$$
which in turn completes our proof.
\qed\enddemo
 =20
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 =20
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 =20
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 =20

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=20
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=20
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=20
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=20

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\Refs
\widestnumber \key{JMP}

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\ref
\key Al
\by D.E. Alspach=20
\paper A Fixed Point Free Nonexpansive Map
\jour Proc. Amer. Math. Soc.
\vol 82 \yr 1981
\pages 423--424
\endref

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\ref=20
\key AP
\by N. Aronszajn, P. Panitchpakdi
 \paper Extensions of uniformly continuous transformations and hyperconvex=
 metric spaces
\jour Pacific J. Math. \vol  6 \yr 1956 \pages 405--439
\endref

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\ref=20
\key Ba
\by J.B. Baillon=20
\paper Nonexpansive mappings and hyperconvex spaces=20
\jour Contemp. Math. \vol 72 \yr 1988 \pages 11--19
\endref

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\ref=20
\key Gr1
\by B. Gr\"unbaum
\paper On some covering and intersection properties in Minkowski spaces
\jour  Pacific J. Math. \vol  9 \yr 1959 \pages 487-494
\endref

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\ref
\key Gr2
\bysame
\paper Some applications of expansion constants
\jour Pacific J. Math. \vol  10 \yr 1960 \pages 193--201
\endref

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\ref
\key Gr3
\bysame
\paper Projection constants
\jour Trans. Amer. Math. Soc. \vol95 \yr 1960 \pages 451--465
\endref

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\ref
\key Ju
\by H.W.E. Jung
\paper \"Uber die kleinste Kugel, die eine r\"aumliche Figur einschlie\ss t
\jour J. Reine Angew. Math. \vol 123 \yr 1901 \pages 241--257
\endref

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\ref
\key Kh
\by M.A. Khamsi
\paper On  Metric Spaces With Uniform  Normal Structure
\jour Proc. Amer. Math. Soc.
\vol 106 \yr 1989
\pages 723--726
\endref

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\ref
\key KLS
\by M.A. Khamsi, M. Lin, R.C. Sine \paper On the fixed points of
commuting nonexpansive maps in hyperconvex spaces
\jour J. Math. Anal. Appl. \vol 168 \yr 1992 \pages 372--380
\endref

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\ref=20
\key  KR
\by M.A. Khamsi, S. Reich=20
\paper Nonexpansive mappings and
semigroups in hyperconvex spaces \jour Math. Japon.\vol 35 \yr 1990 \pages=
 467--471
\endref

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\ref
\key Ki
\by W. Kirk \paperinfo Private communication
\endref

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\ref
\key Kl
\by V.L. Klee Jr.
\paper The support property of a convex set in a linear normed space
\jour Duke Math. J.
\vol 15 \yr 1948
\pages 767--772
\endref

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\ref
\key Lim
\by T.C. Lim=20
\paper Asymptotic Centers and Nonexpansive Mappings in Conjugate
Banach Spaces
\jour Pacific J. Math.
\vol 90\yr 1980
\pages 135--143
\endref

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\ref
\key LS
\by M. Lin, R.C. Sine
\paper Retractions on the fixed point set
of semigroups of nonexpansive maps in hyperconvex spaces \toappear
\endref

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\ref
\key Pe
\by J.P. Penot=20
\paper Fixed point theorems without convexity
\jour Bull. Soc. Math. France
\vol 60
\yr 1979
\pages 129--152
\endref

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\ref
\key Si1
\by  R.C. Sine
\paper Hyperconvexity and approximate fixed points
\jour Nonlinear Anal. \vol 13 \yr 1989\pages 863--869
\endref

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\ref=20
\key Si2
\bysame \paper On linear contraction semigroups in sup
norm spaces \jour Nonlinear Anal. \vol 3 \yr 1979 \pages 885--890
\endref

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\ref=20
\key Soa
\by P. Soardi=20
\paper Existence of fixed points of nonexpansive
mappings in certain Banach lattices \jour Proc. Amer. Math. Soc. \vol73
\yr 1979  \pages 25--29
\endref

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\ref
\key Sob
\by A. Sobczyk=20
\paper Projection of the space $(m)$ on its subspace $(c_0)$=20
\jour Bull. Amer. Math. Soc.=20
\vol 47 \yr 1941 \pages  938--947
\endref
\endRefs

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\enddocument

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\ref =20
\key Is=20
\by J.R. Isbell=20
\paper Six theorems about injective metric spaces=20
\jour Comment. Math. Helvetici \vol 39 \yr 1964 \pages 439--447
\endref

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\ref \key La=20
\by H.E. Lacey \book The isometric theory of classical Banach
spaces  \publ Springer-Berlin, Heidelberg-New York \yr 1974
\endref

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\ref
\key Lin
\by J. Lindenstrauss
\paper On nonlinear projections in Banach spaces
\jour Mich. Math. J.
\vol 11\yr 1964\pages 263--287
\endref

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Theorem~1 establishes that spaces which satisfy Penot's Intersection=
 Property are $2$-hyperconvex. (Thus reflexive Banach spaces and dual Banach=
 spaces are $2$-hyperconvex.) On the other hand, in Section~5, we construct=
 two examples of  $2$-hyperconvex spaces, which fail the Intersection=
 Property.  =20

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Helmut Knaust
The University of Texas at El Paso
Department of Mathematical Sciences
El Paso TX 79968-0514             (915) 747-7002
<http://www.math.utep.edu/Faculty/helmut>

--=====================_940921995==_
Content-Type: text/plain; charset="us-ascii"


Dr. M.A. Khamsi
Department of Mathematical Sciences
The University of Texas at El Paso
El Paso, TX 79968.
--=====================_940921995==_--

