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\title{Some Results on Metric Trees}
\class{Banach Algebras 2009}
\author{Asuman G\"{u}ven Aksoy (Claremont McKenna College) and Timur Oikhberg ( University of California at Irvine)}
\year{2009}
%\email: {aaksoy@cmc.edu}
%\copyrightholder{Asuman G\"{u}ven Aksoy}

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\begin{document}

\begin{poster}

\section{Metric Trees}

Let $M$ be a metric space with metric $d$.


\begin{definition}
 For $x,y \in M$, a \textit{geodesic segment} from $x$ to $y$ is the image of an isometric embedding $\alpha : [a,b] \rightarrow M$ such that $\alpha (a)= x$ and $\alpha (b)= y$. The geodesic segment will be called \emph{metric segment} and is denoted by $[x, y]$.
\end{definition}



\begin{thmbox}
\begin{definition}
We call $M$ a \textit{metric tree} if for any $x,y,z \in M$, the following hold:
\begin{enumerate}
\item there is a unique metric segment between $x$ and $y$.
\item if $[x,z] \cap [z,y] = \{ z \}$, then $[x,z] \cup [z,y] = [x,y]$.
\end{enumerate}
\end{definition}
\end{thmbox}

The following is an example of metric tree.
\begin{example}
Let $\rho$ denote the Euclidean metric on $\mathbb{R}^2$. We define the \textit{radial metric} by
$$ d(x,y) =
\begin{cases}
\rho(x,y) & \text{ if } y = tx \text{ for some } t \in \mathbb{R} \\
\rho(x,0) + \rho(y,0) & \text{otherwise}.
\end{cases} $$

\end{example}


\begin{definition}
Let $M$ be a metric tree, and let $A \subseteq M$. We call $ F_A = \left\{ f \in A : f \notin (x,y) \text{ for all } x,y \in A \right\} $ the set of \textit{final points} of $A$. Here, $(x,y) = [x,y] \backslash \{ x,y \}$.
\end{definition}


This concept leads us to a characterization of compact metric trees.

\begin{thmbox}
\begin{theorem}(Aksoy, Borman, Westfahl)
A metric tree $(M,d)$ is compact if and only if $$ M = \bigcup_{f \in F_M}{[a,f]} \text{ for all } a \in M, \text{ and } \overline{F}_M \text{ is compact.} $$
\end{theorem}
\end{thmbox}
\section{Hyperconvexity}
\begin{definition}
A metric space $M$ is called \emph{hyperconvex} if $\displaystyle \bigcap_{i\in I} B_c (x_i,r_i) \neq \emptyset$ for any collection $\{B_c(x_i;r_i)\}_{i \in I}$ of closed balls in $X$ with $x_ix_j \leq r_i + r_j$.
\end{definition}
\begin{thmbox}
\begin{theorem}(Aronszajn and Panitchpakdi)
    $X$ is a hyperconvex metric space if and only if
    for all metric spaces $D$, if $C \subset D$ and $f: C \to X$ is a nonexpansive
    mapping, then $f$ can be extended to the nonexpansive mapping
    $\tilde{f}:D \to X$.
\end{theorem}
\end{thmbox}
The simplest example of hyperconvex space is the set of real numbers $\mathbb{R}$ or a finite dimensional real Banach space endowed with the maximum norm. While
the Hilbert space $\ell^2$ fails to be hyperconvex, the spaces
$\ell^{\infty}$ and $L^{\infty}$ are hyperconvex.
The connection between hyperconvex metric spaces and metric trees is given in the following theorem:
\begin{thmbox}

\begin{theorem}(Aksoy and Kirk)
    A complete metric tree $T$ is hyperconvex.
\end{theorem}
\end{thmbox}
\section{Embeddings of Metric Trees}

\begin{definition}

    Let $X$ be a metric space.  Define $\ell^\infty(X)$, as the
    normed linear space where
    $$
    \ell^\infty(X):= \left\{(x_m)_{m\in X} \mid x_m \in \mathbb{R},
    \sup_{m \in X} |x_m|< \infty \right\}$$
     and $ \|(x_m)\| := \sup_{m\in X}|x_m|$.
  \end{definition}

\begin{thmbox}
\begin{theorem}(Kirk and Sims)
Let $X$ be a metric space and $a \in X$, then
    \[
    J: X \to \ell^\infty(X) \text{ where } J(x) = (xm - am)_{m\in X}
    \]
    is an isometric embedding of $X$ into $\ell^\infty(X)$.
\end{theorem}
\end{thmbox}

We define a ``canonical" embedding $J = J_{x^*}$ of tree $T$
into $\ell_{\infty} (T)$ ($x^*$ is a point in $T$) by
$$
J_{x^*}(x)(y) = J(x)(y) = d(x,y) - d (x^*,y) .
$$
When $T$ is finite, we can also use the embedding $$J(x)(y)= d(x,y).$$
We also use the ``semicanonical'' embedding of $T$ into $\ell_1$ (for such embedding see ~Section 2.5 of S. Evans, {\em Probability and real trees}).
We have not been able to construct explicit embeddings of metric trees into
other Banach spaces. However, we can show:
\begin{thmbox}
\begin{theorem}
Suppose $X$ is a superreflexive Banach space, $T$ is a finitely
generated metric tree, and $\varepsilon > 0$. Then there exists a
Banach space $Y$, $(1+\varepsilon)$-isomorphic to $X$, such that $T$
embeds into $Y$ isometrically.
\end{theorem}
\end{thmbox}
\section{Barycenters in Metric Trees}

Suppose $U$ is an isometric embedding of a metric tree $T$
into a Banach space $X$, equipped with the norm $\| \cdot \|$.
Suppose $x_1, \ldots, x_n \in T$, and let $\tlx = (x_1 + \ldots + x_n)/n$
be their barycenter in $X$ (we identify $x\in T$ with $U(x) \in X$). Let
$\mathbf{P} = \mathbf{P}_{U,T,X}$ be the set of contractive retractions $\pi$ from $X$ onto $U(T)$
(it is non-empty since $T$ is injective). We try to describe $\mathbf{P}(\tlx)$. More generally, suppose
$\alpha = (\alpha_i)_{i=1}^n$ is a sequence of positive numbers, with
$\sum_{k=1}^n \alpha_k = 1$. Set $\tla = \sum_{k=1}^n \alpha_k x_k$,
and try to describe $\mathbf{P}(\tla)$.

\begin{thmbox}
\begin{theorem}
Suppose $T$ is a complete metric tree, embedded into $\ell_\infty(T)$
in the canonical way. For $x_0 \in T$, the following are equivalent:
\begin{enumerate}
\item
$x_0 \in \pp(\tla)$.
\item
$\|x_0 - x\| \leq \sum_k \alpha_k \|x_k - x\|$ for any $x \in T$.
\end{enumerate}
\end{theorem}
\end{thmbox}
\begin{thmbox}
\begin{theorem}
Suppose $T$ is a finitely generated tree, embedded into $\ell_1$
in the semicanonical way. For $x_0 \in T$, the following are equivalent:
\begin{enumerate}
\item
$x_0 \in \pp(\tla)$.
\item
$\|x_0 - x\| \leq \sum_k \alpha_k \|x_k - x\|$ for any $x \in T$.
\end{enumerate}
\end{theorem}
\end{thmbox}
\begin{thmbox}
\begin{proposition}
Suppose a complete metric tree $T$ is embedded isometrically into a normed space $X$,
and $\tilde{x}$ is a point of $X$. Then $\pp(\tilde{x})$ is a closed, metric convex
subset of $T$.
\end{proposition}
\end{thmbox}

\section{Type and Cotype}
We say that a metric space $(X,d)$ satisfies {\it the four-point inequality}
if, for any $x_1, x_2, x_3, x_4 \in X$,

$d(x_1,x_2) + d(x_3,x_4) \leq $ \\$\max\{ d(x_1,x_3) + d(x_2,x_4) ,
d(x_1,x_4) + d(x_2,x_3) \}$.

\begin{thmbox}
\begin{theorem}For type and cotype of metric trees we have:
\begin{enumerate}
\item
Any metric space satisfying the four-point inequality has metric type 2, with
constant $1$. In particular, this result holds for metric trees.
\item
Any complete metric tree has metric cotype 2, with a universal constant.
\end{enumerate}
\end{theorem}
\end{thmbox}
\section{Entropy Quantities}
\begin{definition}
 In the following, for $\ep$-cover sets of diameter $ \leq 2 \ep$ and for $\ep$-net balls of radius $\ep$ is used.
\begin{enumerate}

\item  Let $\mathcal{N}_\ep(A)$ be the cardinality of a minimal $\ep$-cover of $A$.
    Then define the \emph{$\ep$-entropy of $A$} as
    $$\mathcal{H}_\ep(A):= \log_2 \mathcal{N}_\ep(A).$$

    Similarly, let $\mathcal{N}_\ep^{M}(A)$ be the cardinality of a minimal $\ep$-net
    for $A$ in $M$.     Then define the \emph{$\ep$-entropy of $A$ relative to $M$} as
    $$\mathcal{H}_\ep^{M}(A):= \log_2 \mathcal{N}_\ep^{M}(A).$$
\item $A$ is \emph{centered} if for all $U\subset A$ such that $\diam(U) =2r$, there exists $a\in A$ such that $U \subset B_{c} (a;r)$.
\item Given a normed linear space $X$ and subset $A$, define the
    \emph{nth Kolmogorov diameter ($n$-width) of $A$ in $X$} as:
    \[
 \delta_n(A):= \inf \left\{ \sup_{a \in A} \, d(a, M_n) \mid M_n
        \text{ is a $n$-dim subsp. of }X \right\}.
    \]
\item Let $T$ be a metric tree and $A \subset T$, then define the
    \emph{nth Kolmogorov diameter of $A$ in $\ell^\infty(T)$} as:
    \[
    \delta_n(A):= \delta_n(J(A), \ell^\infty(T)).
    \]

\end{enumerate}
Recall that for a metric tree $T$, we have the isometric embedding $J: T \to \ell^\infty(T)$.  If $T$ is a complete metric tree, then $T$ is hyperconvex and thus injective so we have a nonexpansive projection $P: \ell^\infty(T) \to T$, and where $P \circ J = \id_T$.


\end{definition}

\begin{thmbox}
\begin{theorem}

For entropy quantities and other measures of noncompactness in metric trees we have:

\begin{enumerate}
\item Every complete metric tree $T$ is centered.
\item For a complete metric tree $T$, if $A \subset T$, then
$$\mathcal{H}_\ep^{T}(A) = \mathcal{H}_\ep(A). $$
\item Suppose $S$ is a compact subset of a complete metric tree $T$, and $\varepsilon_1, \varepsilon_2$
are positive numbers. Then
% $$
% N(\con(S),\vr_1+\vr_2) \leq N(S,\vr_1) \Big\lceil \frac{\diam S}{4\vr_2} \Big\rceil .
% $$
$$\mathcal{N}_{\varepsilon_1+\varepsilon_2}(\mbox{con}(S)) \leq \mathcal{N}_{\varepsilon_1}(S) \lceil \diam S/(4\varepsilon_2) \rceil.$$
\item If $T$ is a complete metric tree, where $A \subset T$ bounded, then
    \[
    \lim_{n \to \infty} \delta_n(A) = \beta(A).
    \]


Where  $\beta(A)$ is the Hausdorff measure of noncompactness defined as \[
    \beta (A) :=\inf\,\left\{b>0 \mid A \subset \bigcup_{j=1}^{n} B(x_j;b)
    \text{ for some } x_j \in T\right\}.
    \]

    \end{enumerate}
\end{theorem}
\end{thmbox}

\section{Acknowledgments}

The first named author thanks the European Science Foundation (ESF-EMS-ERCOM partnership) for conference support.
E-mail: aaksoy@cmc.edu

%\begin{thebibliography}{1}

%\bibitem{abw} A. G. Aksoy, M. S. Borman, and A. L. Westfahl, {\em Compactness and measures of noncompactness in metric trees}. Proc. Intl. Symp. on Banach and Function Spaces II, Kitakyushu, Japan, 277--292, Yokohama, 2008.

%\bibitem{blumenthal} L.M. Blumenthal, {\em Theory and applications of distance geometry}, Second Edition, Chelsea Publishing Co., New York 1970.

%\end{thebibliography}

%\section{For Further Information}

\end{poster}
\end{document} 