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A Note on Geodesically Bounded
-Trees
Fixed Point Theory and Applications volume 2010, Article number: 393470 (2010)
Abstract
It is proved that a complete geodesically bounded -tree is the closed convex hull of the set of its extreme points. It is also noted that if
is a closed convex geodesically bounded subset of a complete
-tree
and if a nonexpansive mapping
satisfies
then
has a fixed point. The latter result fails if
is only continuous.
1. Introduction
Recall that for a metric space a geodesic path (or metric segment) joining
and
in
is a mapping
of a closed interval
into
such that
and
for each
Thus
is an isometry and
An
-tree (or metric tree) is a metric space
such that:
(i)there is a unique geodesic path (denoted by ) joining each pair of points
(ii)if then
From (i) and (ii), it is easy to deduce that
(iii)if then
for some
The concept of an -tree goes back to a 1977 article of Tits [1]. Complete
-trees posses fascinating geometric and topological properties. Standard examples of
-trees include the "radial" and "river" metrics on
For the radial metric, consider all rays emanating from the origin in
Define the radial distance
between
to be the usual distance if they are on the same ray; otherwise take
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F393470/MediaObjects/13663_2010_Article_1273_Equ1_HTML.gif)
(Here denotes the usual Euclidean distance and
denotes the origin.) For the river metric
on
, if two points
, and
are on the same vertical line, define
Otherwise define
where
and
More subtle examples of
-trees also exist, for example, the real tree of Dress and Terhalle [2].
It is shown in [3] that -trees are hyperconvex metric spaces (a fact that also follows from Theorem B of [4] and the characterization of [5]). They are also CAT
spaces in the sense of Gromov (see, e.g., [6, page 167]). Moreover, complete and geodesically bounded
-trees have the fixed point property for continuous maps. This fact is a consequences of a result of Young [7] (see also [8]), and it suggests that complete geodesically bounded
-trees have properties that one often associates with compactness. The two observations below serve to affirm this.
2. A Krein-Milman Theorem
In [9] Niculescu proved that a nonempty compact convex subset of a complete CAT
space (called a global NPC space in [9]) is the convex hull of the set of all its extreme points. Subsequently, in [10], Borkowski et al. proved (among other things) that compactness is not needed in the special case when
is a complete and bounded
-tree. Here we show that in complete
-trees even the boundedness assumption may be relaxed.
Theorem 2.1.
Let be a complete and geodesically bounded
-tree. Then
is the convex hull of its set
of extreme points.
Proof.
Let and let
We will show that
lies on a segment joining
to some other element of
We proceed by transfinite induction. Let
denote the set of all countable ordinals, let
let
and assume that for all
with
has been defined so that the following condition holds:
(i) and
There are two cases.
(1) If
there is nothing to prove because
. Otherwise, there are elements
such that
lies on the segment
and
At least one of these points
say
does not lie on the segment
. Set
and observe that
lies on the segment
(2) is a limit ordinal. Since
is geodesically bounded, it must be the case that
This implies that
is a Cauchy net. Since
is complete, it must converge to some
Therefore, is defined for all
Since
is geodesically bounded,
But since
is uncountable, it is not possible that
for each
Hence this transfinite process must terminate, and
for some
It now follows from (i) that
and
lies on the segment
Remark 2.2.
The above proof shows that in fact each point of is on a segment joining any given extreme point to some other extreme point.
3. A Fixed Point Theorem
It is known that if is a bounded closed convex subset of a complete CAT
space
and if
is a nonexpansive mapping for which
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F393470/MediaObjects/13663_2010_Article_1273_Equ2_HTML.gif)
then has a fixed point (see [11, Theorem
]
also [12, Corollary
]). This fact carries over to
-trees since
-trees are also CAT
spaces. However, we note here that if
is an
-tree, then again boundedness of
can be replaced by the assumption that
is merely geodesically bounded. In fact, we prove the following. (In the following theorem, we assume
is nonexpansive relative to the Hausdorff metric on the bounded nonempty closed subsets of
Theorem 3.1.
Suppose is a closed convex and geodesically bounded subset of a complete
-tree
and suppose
is a nonexpansive mapping taking values in the family of nonempty bounded closed convex subsets of
Suppose also that
Then there is a point
for which
We will need the following result in the proof of Theorem 3.1. (See [13, 14] for more general set-valued versions of this theorem.)
Theorem 3.2.
Suppose is a closed convex geodesically bounded subset of a complete
-tree
and suppose
is continuous. Then either
has a fixed point or there exists a point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F393470/MediaObjects/13663_2010_Article_1273_Equ3_HTML.gif)
Proof of Theorem 3.1.
Since complete -trees are hyperconvex, by Corollary
of [15] the selection
defined by taking
to be the point of
which is nearest to
for each
is a nonexpansive single-valued mapping. Now assume
does not have a fixed point. Then by Theorem 3.2 there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F393470/MediaObjects/13663_2010_Article_1273_Equ4_HTML.gif)
We assert that for each
Indeed let
By (iii) there exists
such that
But since
is convex
so
implies
Also
so it follows from (3.3) that
Thus
and the segment
must pass through
. Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F393470/MediaObjects/13663_2010_Article_1273_Equ5_HTML.gif)
Thus – a contradiction. Therefore, there exists
such that
Corollary 3.3.
Suppose is a closed convex and geodesically bounded subset of a complete
-tree
and suppose
is a nonexpansive mapping for which
Then
has a fixed point.
Example 3.4.
In view of the fact that continuous self-maps of have fixed points, it is natural to ask whether Corollary 3.3 holds for continuous mappings. The answer is no, even when
is bounded. Let
be the Euclidean plane
with the radial metric. Let
be a sequence of distinct points on the unit circle, and let
We now define a continuous fixed-point free map
for which
. First move each point of the segment
to the right onto a segment
where
and
is on the ray which extends
(Thus
For each
let
denote the point on the segment
which has distance
from
. It is now clearly possible to construct a continuous (even lipschitzian) fixed point-free map
(a shift) of the segment
onto the segment
for which
Thus
for all
Remark 3.5.
Corollary 3.3 for bounded is also a consequence of Theorem
of [15].
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Kirk, W. A Note on Geodesically Bounded -Trees.
Fixed Point Theory Appl 2010, 393470 (2010). https://doi.org/10.1155/2010/393470
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DOI: https://doi.org/10.1155/2010/393470