Somerecentresultsinmetricﬁxedpointtheory · pair of vertices (the edges of ). A comparison triangle for the geodesic triangle (x 1,x 2,x 3)in(X,d) is a triangle (x 1,x 2,x 3):=

J. fixed point theory appl. 2 (2007), 195–207c© 2007 Birkhauser Verlag Basel/Switzerland1661-7738/020195-13, published online 10.10.2007

DOI 10.1007/s11784-007-0031-8

Journal of Fixed Point Theoryand Applications

Some recent results in metric fixed point theory

W. A. Kirk

Dedicated to Edward Fadell and Albrecht Dold on the occasion of their 80th birthdays

Abstract. This is a survey of recent results on best approximation and fixedpoint theory in certain geodesic spaces. Some of these results are related tofundamental fixed point theorems in topology that have been known for manyyears. However the metric approach is emphasized here.

Mathematics Subject Classification (2000). Primary 05C12, 54H25, 47H09.

Keywords. Best approximation, fixed points, CAT(0) spaces, metric trees,hyperconvex spaces.

1. Introduction

Fixed point theory for nonexpansive and related mappings has played a funda-mental role in many aspects of nonlinear functional analysis for many years. Thetheory has traditionally involved an intertwining of geometrical and topologicalarguments in a Banach space setting. However, because the theory is fundamen-tally metric in nature, there has been a trend in recent years to seek applicationsin settings where the underlying algebraic linear structure of a Banach space isnot present. This trend perhaps has its origins in the concept of a hyperconvexmetric space due to Aronszajn and Panitchpakdi [5]. (Definitions of terms dis-cussed in the Introduction will be given later.) The fact that bounded hyperconvexspaces have the fixed point property for nonexpansive mappings has been knownfor many years and is basically due, independently, to Sine [40] and Soardi [42].Subsequently J.-B. Baillon [6] extended this result to commuting families of non-expansive mappings, and since then a flourishing theory has evolved, as evidencedby the Espınola–Khamsi survey in [16].

More recently, many of the standard ideas of nonlinear analysis have beenextended to the class of so-called CAT(0) spaces (see [24], [25]). While many ofthe Banach space ideas carry over to a complete CAT(0) setting without essential

196 W. A. Kirk JFPTA

change, often a more geometrical approach is required, with less emphasis on topo-logical concepts caused by, among other things, the absence of a weak topology.

There is an interesting class of spaces which are both complete CAT(0) spacesand hyperconvex metric spaces. These are the complete R-trees (or metric trees).Indeed, a CAT(0) space is hyperconvex if and only if it is a complete R-tree.

In this survey we discuss some recent metric fixed point results in each ofthe settings just described, with emphasis on results in R-trees. In fact, the resultsin R-trees have some interesting connections with classical fixed point results intopology.

We begin by describing each of these settings in more detail. We stress themetric approach here, although some of these results may be derived from moreabstract theory; see, e.g., Horvath [20].

2. Preliminaries

Hyperconvex spaces

A metric space Y is said to be hyperconvex if every family {B(yα; rα)}α∈A of closedballs centered at yα ∈ Y with radii rα ≥ 0 has nonempty intersection whenever

d(yα, yβ) ≤ rα + rβ ∀α, β ∈ A.

Such spaces include, among others, the classical L∞ spaces ([29]). It is known thatcompact hyperconvex spaces (often called Helly spaces) are contractible and locallycontractible; hence they have the fixed point property for continuous mappings (see[35]).

We now list some other properties of hyperconvex spaces.

1. Hyperconvex spaces are metrically convex in the sense of Menger. Thismeans that given any two points x and y in a hyperconvex space M there is apoint z ∈ M, x �= z �= y, such that d(x, z) + d(z, y) = d(x, y).

2. Hyperconvex spaces are complete. This is an easy consequence of the def-inition.

As a consequence of a classical theorem of K. Menger, there is a metricsegment (an isometric image of a real line interval) joining any two points of ahyperconvex space whose length is equal to the distance between the points. Inview of 1 and 2 the following are equivalent: (i) M is hyperconvex; (ii) M ismetrically convex and has the binary ball intersection property, that is, any familyof closed balls in M has nonempty intersection whenever any two of its membersintersect.

3. Hyperconvex spaces are injective [5]. This means that if M is hyperconvexand if f : X → M is a nonexpansive mapping defined on a metric space X, thenf has a nonexpansive extension f : Y → M to any metric space Y ⊃ X.

4. If M is hyperconvex, and if M is a subspace of a metric space Y, thenthere is a nonexpansive retraction of Y onto M. This follows from 3.

Vol. 2 (2007) Recent results in metric fixed point theory 197

5. Every metric space M has a hyperconvex hull ; that is, there is a hyper-convex metric space ε(M) which contains an isometric copy of M, and whichhas the property that no proper subset of ε(M) which contains M metrically ishyperconvex. This is a result of Isbell [21].

6. ([42], [40], [6]) If M is a bounded hyperconvex metric space, and if f :M → M is a nonexpansive mapping, then f has a fixed point. In fact, a commutingfamily of nonexpansive mappings of M → M always has a common fixed point.

7. ([22], [28]) If M is a compact hyperconvex metric space and if f : M → Mis continuous (or condensing), then f has a fixed point.

A discussion of the preceding facts and many more can be found in thesurvey [16].

R-trees (metric trees)

There are many equivalent definitions of R-tree. Here are two of them.

Definition 1. An R-tree is a metric space M such that for every x and y in Mthere is a unique arc between x and y and this arc is isometric to an interval in R

(i.e., is a geodesic segment).

Definition 2. An R-tree is a metric space M such that(i) there is a unique geodesic segment denoted by [x, y] joining each pair of points

x and y in M ;(ii) [y, x] ∩ [x, z] = {x} ⇒ [y, x] ∪ [x, z] = [y, z].

The following is an immediate consequence of (ii).(iii) If x, y, z ∈ M there exists a point w ∈ M such that [x, y] ∩ [x, z] = [x,w]

(whence by (i), [x,w] ∩ [z, w] = {w}).Standard examples of R-trees include the “radial” and “river” metrics on R

2.For the radial metric, consider all rays emanating from the origin in R

2. Definethe radial distance dr between x, y ∈ R

2 as follows:

dr(x, y) = d(x, 0) + d(0, y).

(Here d denotes the usual Euclidean distance and 0 denotes the origin.) For theriver metric ρ, if two points x, y are on the same vertical line, define ρ(x, y) =d(x, y). Otherwise define ρ(x, y) = |x2| + |y2|+ |x1 − y1|, where x = (x1, x2) andy = (y1, y2).

Much more subtle examples exist; e.g., the real tree of Dress and Terhalle [15].The concept of an R-tree goes back to a 1977 article of J. Tits [43]. The idea

has also been attributed to A. Dress [13], who first studied the concept in 1984and called it T -theory.

Bestvina [8] observes that much of the importance of R-trees stems from thefact that in many situations a sequence of negatively curved objects (manifolds,groups) gives rise (in some sense “converges”) to an R-tree together with a groupacting on it by isometries. There are applications in biology and computer science


as well. The relationship with biology stems from the construction of phylogenetictrees [39]. Concepts of “string matching” are also closely related with the structureof R-trees [7].

CAT(0) spaces

A metric space is a CAT(0) space (the term is due to M. Gromov—see, e.g., [9,p. 159]) if it is geodesically connected, and if every geodesic triangle in X is at leastas “thin” as its comparison triangle in the Euclidean plane. The precise definitionis given below. For a thorough discussion of these spaces and of the fundamentalrole they play in various branches of mathematics, see Bridson and Haefliger [9]or Burago et al. [11]. We note in particular that a complex Hilbert ball with thehyperbolic metric (see [19]; also inequality (4.3) of [38] and subsequent comments)is a CAT(0) space.

Let (X, d) be a metric space. A geodesic path joining x ∈ X to y ∈ X (or,more briefly, a geodesic from x to y) is a map c from a closed interval [0, l] ⊂ R

to X such that c(0) = x, c(l) = y, and d(c(t), c(t′)) = |t − t′| for all t, t′ ∈ [0, l]. Inparticular, c is an isometry and d(x, y) = l. The image α of c is called a geodesic(or metric) segment joining x and y. When it is unique this geodesic is denoted[x, y]. The space (X, d) is said to be a geodesic space if any two points of X arejoined by a geodesic, and X is said to be uniquely geodesic if there is exactly onegeodesic joining x and y for each x, y ∈ X. A subset Y ⊆ X is said to be convexif Y includes every geodesic segment joining any two of its points.

A geodesic triangle �(x1, x2, x3) in a geodesic metric space (X, d) consistsof three points in X (the vertices of �) and a geodesic segment between eachpair of vertices (the edges of �). A comparison triangle for the geodesic triangle�(x1, x2, x3) in (X, d) is a triangle �(x1, x2, x3) := �(x1, x2, x3) in the Euclideanplane E

2 such that dE2(xi, xj) = d(xi, xj) for i, j ∈ {1, 2, 3}.A geodesic metric space is said to be a CAT(0) space if all geodesic triangles

of appropriate size satisfy the following comparison axiom.Let � be a geodesic triangle in X and let � be a comparison triangle for �.

Then � is said to satisfy the CAT(0) inequality if for all x, y ∈ � and all compar-ison points x, y ∈ �, d(x, y) ≤ dE2(x, y).

The following theorem yields a characterization of hyperconvex CAT(0)spaces.

Theorem 1 ([23]). For a metric space M the following are equivalent:

(i) M is a complete R-tree;(ii) M is hyperconvex and has unique metric segments.

It is known that a complete R-tree is a complete CAT(0) space ([9, p. 167]).On the other hand, a CAT(0) space has unique metric segments. If it is alsohyperconvex then by Theorem 1 it must be a complete R-tree. Thus we have:

Theorem 2. A CAT(0) space is hyperconvex if and only if it is a complete R-tree.


A proof that a complete R-tree is injective is given in [30]. Since injectivespaces are known to be hyperconvex ([5]), this also gives (i)⇒(ii). Another proofthat (i)⇒(ii) is given in Aksoy and Maurizi [4]. Their proof is based on an inter-esting four-point property of metric trees.

Definition 3. A metric space (X, d) is said to satisfy the four-point property if foreach set of four points x, y, z, w ∈ X the following holds:

d(x, y) + d(u,w) ≤ max{d(x, u) + d(y, w), d(x,w) + d(y, u)}.Since one obtains the triangle inequality by taking u = v, the four-point

property is a stronger condition. In [13] it is shown that a metric space is a completeR-tree if and only if it is complete, connected, and satisfies the four-point property.

We now turn to the Lifshits character of these spaces. Balls in X are said tobe c-regular if the following holds: For each k < c there exist μ, α ∈ (0, 1) suchthat for each x, y ∈ X and r > 0 with d(x, y) ≥ (1 − μ)r, there exists z ∈ X suchthat

B(x; (1 + μ)r) ∩ B(y; k(1 + μ)r) ⊂ B(z;αr). (2.1)The Lifshits character κ(X) of X is defined as follows:

κ(X) = sup{c ≥ 1 : balls in X are c-regular}.A mapping f : X → X is said to be eventually k-lipschitzian if there exists

n0 ∈ N such that d(fn(x), fn(y)) ≤ kd(x, y) for all x, y ∈ X and n ≥ n0. The Lif-shits character is fundamental in metric fixed point theory because of the followingresult.

Theorem 3 (Lifshits [31]). Let (X, d) be a complete metric space. Then every even-tually k-lipschitzian mapping T : X → X with k < κ(X) has a fixed point if it hasa bounded orbit.

The Lifshits character is known for many classical Banach spaces. For aHilbert space it is

√2. The following is proved in [12].

Theorem 4. If (X, d) is a complete CAT(0) space, then κ(X) ≥√

2. Moreover, ifX is an R-tree, then κ(X) = 2.

Another proof of the second statement is given in [1, Theorem 3.16]; also acharacterization of compact R-trees in terms of metric segments is found there.

In view of Theorem 3, if X is a complete bounded CAT(0) space then everyeventually k-lipschitzian mapping T : X → X with k <

√2 has a fixed point. The

corresponding fact for a complete R-tree is the following.

Theorem 5. Let X be a complete R-tree and let T : X → X be eventually uniformlyk-lipschitzian for k < 2, and assume that T has bounded orbits. Then T has a fixedpoint.

For a direct proof of this result (and related facts), see [3]. The significance ofthe above result lies in the fact that the mapping is not assumed to be continuous.Cf. Theorem 7 below.


Gated sets

Many of the ideas discussed above, especially those in R-trees, can be couched ina more abstract framework. A subset Y of a metric space X is said to be gated([14]) if for any point x /∈ Y there exists a unique point xY ∈ Y (called the gateof x in Y ) such that for any z ∈ Y,

d(x, z) = d(x, xY ) + d(xY , z).

Obviously gated sets in a complete geodesic space are always closed and convex.It is known ([14]) that gated subsets of a complete geodesic space X are

proximinal nonexpansive retracts of X. Specifically, if A is a gated subset of X,then the mapping that associates with each point x in X its gate in A (i.e., thegate map, or “nearest point map”) is nonexpansive. Several other properties ofgated sets can be found, for example, in [44, p. 98]) In particular it can be easilyshown by induction that the family of gated sets in a complete geodesic space Xhas the Helly property. Thus if S1, . . . , Sn is a collection of pairwise intersectinggated sets in X then

⋂ni=1 Si �= ∅.

The gated subsets of an R-tree are precisely its closed and convex subsets.Thus the following results apply to R-trees.

Proposition 1 ([17]). Let (X, d) be a complete geodesic space, and let {Hα}α∈Λ

be a collection of nonempty gated subsets of X which is downward directed by setinclusion. If X (or more generally, some Hα) does not contain a geodesic ray, then⋂

α∈Λ Hα �= ∅.

Proposition 2 ([17]). Let (X, d) be a complete geodesic space, and let {Hn} bea descending sequence of nonempty gated subsets of X. If {Hn} has a boundedselection, then

⋂∞n=1 Hn �= ∅.

3. The fixed point property for R-trees

G. S. Young, Jr. obtained the following result in 1946. He notes explicitly in [47]that compactness is not needed.

Theorem 6 ([46]). Let M be an arcwise connected Hausdorff space which is suchthat every monotone increasing sequence of arcs is contained in an arc. Then Mhas the fixed point property (for continuous maps).

In [33], J. C. Mayer and L. G. Oversteegen proved that for a separable metricspace (X, d) the following are equivalent:

1. X is an R-tree.2. X is a locally arcwise connected and uniquely arcwise connected metric space.

If a complete R-tree is geodesically bounded it is easy to see that everymonotone increasing sequence of arcs is contained in an arc. In view of this, wehave the following.


Theorem 7. A complete geodesically bounded R-tree has the fixed point propertyfor continuous maps.

Although the validity of Theorem 7 goes back to Young’s 1946 result, a moreconstructive metric approach might be of interest. The following proof is takenfrom [26].

Proof of Theorem 7. For u, v ∈ X we let [u, v] denote the (unique) metric segmentjoining u and v and let [u, v) = [u, v]\{v}. We associate with each point x ∈ X apoint ϕ(x) as follows. For each t ∈ [x, f(x)], let ξ(t) be the point of X for which

[x, f(x)] ∩ [x, f(t)] = [x, ξ(t)].

(It follows from the definition of an R-tree that such a point always exists.) Ifξ(f(x)) = f(x) take ϕ(x) = f(x). Otherwise it must be the case that ξ(f(x)) ∈[x, f(x)). Let

A = {t ∈ [x, f(x)] : ξ(t) ∈ [x, t]}, B = {t ∈ [x, f(x)] : ξ(t) ∈ [t, f(x)]}.

Clearly A ∪ B = [x, f(x)]. Since ξ is continuous, both A and B are closed. AlsoA �= ∅ as f(x) ∈ A. However, the fact that f(t) → f(x) as t → x implies B �= ∅(because t ∈ A implies d(f(t), f(x)) ≥ d(t, x).) Therefore there exists a pointϕ(x) ∈ A ∩ B. If ϕ(x) = x then f(x) = x and we are done. Otherwise x �= ϕ(x)and

[x, f(x)] ∩ [x, f(ϕ(x))] = [x, ϕ(x)].

Now let x0 ∈ X, and let xn = ϕn(x0). Assuming the process does not termi-nate upon reaching a fixed point of f , by construction the points {x0, x1, x2, . . .}are linear and thus lie on a subset of X which is isometric with a subset of thereal line, i.e., on a geodesic. Since X does not contain a geodesic of infinite lengthit must be the case that

∞∑

i=0

d(xi, xi+1) < ∞,

and hence that {xn} is a Cauchy sequence. Suppose limn→∞ xn = z. Then bycontinuity

limn→∞

f(xn) = f(z),

and in particular {f(xn)} is a Cauchy sequence. However, by construction,

d(f(xn), f(xn+1)) = d(f(xn), xn+1) + d(xn+1, f(xn+1)).

Since limn→∞ d(f(xn), f(xn+1)) = 0 it follows that limn→∞ d(f(xn), xn+1) =d(f(z), z) = 0 and f(z) = z. �


4. Best approximation in R-trees

Ky Fan’s classical best approximation theorem (see [18]) asserts that if C is anonempty compact convex subset of a normed linear space E and if f : C → E iscontinuous, then there exists a point z ∈ C such that

‖z − f(z)‖ = inf{‖x − f(z)‖ : x ∈ C}.Over the years this theorem has been extended in various ways; see e.g., Singh etal. [41] for a discussion. Also [36] and [37] provide extensions of Fan’s theorem toset-valued mappings and to noncompact sets, respectively.

There have been two recent approaches to best approximation for set-valuedmappings in R-trees. In [27] Fan’s best approximation theorem is extended to uppersemicontinuous mappings in an R-tree. The proof given in [27] is constructive—amodification of the proof of Theorem 7—although as we note below there is a nicetopological approach. A second approach is found in [32] where it is shown that alower semicontinuity assumption also suffices.

We begin with the approach of [27]. Once again we assume that the space Xis geodesically bounded, that is, we assume that X does not contain a geodesic ofinfinite length.

Theorem 8 ([27]). Suppose X is a closed convex subset of a complete R-tree Y , andsuppose X is geodesically bounded. Let T : X → 2Y be an upper semicontinuousmapping whose values are nonempty closed convex subsets of Y . Then there existsa point z ∈ X such that

dist(z, T (z)) = infx∈X

dist(x, T (z)).

Let X be a connected Hausdorff space. A point p ∈ X separates u, v ∈ X ifu and v are contained in disjoint open subsets of X\{p}. If e ∈ X it is possible todefine a relation Γe on X × X in the following way:

Γe = ({e × X}) ∪ Δ(X × X) ∪ {(x, y) : x separates e from y}.It is known (see Ward [45]) that Γe is a partial order.

A connected Hausdorff space X is said to satisfy property D(3) if the followingcondition holds: If A and B are disjoint closed connected subsets of X, then thereexists z ∈ X such that z separates A and B.

L. E. Ward, Jr. obtained the following result in 1974.

Theorem 9 ([45]). Suppose X is a connected Hausdorff space that satisfies propertyD(3). Suppose also that there exists e ∈ X such that, relative to Γe, each chain inX has a maximal element and a minimal element. Let f : X → 2X be an uppersemicontinuous mapping whose values are nonempty closed connected subsets of X.Then f has a fixed point.

Theorem 8 is an easy consequence of Ward’s theorem.

Proof of Theorem 8 ([27]). It is known that the nearest point projection of aCAT(0) space onto a closed convex subset of the space is nonexpansive ([9, p. 176]),


and since an R-tree is a CAT(0) space ([9, p. 167]), the nearest point map R of Yonto X is nonexpansive. Hence the map R ◦ T : X → 2X is upper semicontinuousand has a fixed point z by Theorem 9. Thus there exists y ∈ T (z) such thatR(y) = z. However, since R is the nearest point map, it must be the case thatR(y) = z for all y ∈ T (z). If z ∈ T (z) we are finished. Otherwise, choose y1 ∈ T (z)such that d(z, y1) = dist(z, T (z)). Then if x ∈ X and x �= z,

dist(z, T (z)) = d(z, y1) < d(x, z) + d(z, y1) = dist(x, T (z)). �In fact, the following extension of Theorem 9 actually gives a topological

version of Fan’s best approximation theorem. In this theorem xΓe := {z ∈ X :x ≤ z} where x ≤ z means (x, z) ∈ Γe.

Theorem 10 ([27]). Suppose Y is a connected Hausdorff space that satisfies propertyD(3) and suppose X is a closed and connected subset of Y. Suppose also that thereexists e ∈ X such that, relative to Γe, each chain in X has a maximal element anda minimal element. Let T : X → 2Y be an upper semicontinuous mapping whosevalues are nonempty closed connected subsets of Y. Then either T has a fixed point,or there exists x ∈ ∂X such that T (x) ⊂ xΓe\{x}.

The following KKM principle for trees is also proved in [27]. It can also beused to give yet another proof of Theorem 8.

Theorem 11 ([27]). Suppose X is a closed convex subset of a complete R-tree H,and suppose G : X → 2H\{∅} has nonempty closed values. Suppose also that foreach finite F ⊂ X,

convH(F ) ⊂⋃

x∈F

G(x).

Then {G(x)}x∈X has the finite intersection property. Moreover, if X is geodesicallybounded, then

⋂x∈X G(x) �= ∅.

We now turn to the results of Markin [32]. For a subset B of a metric space M,define Nε(B) = {x ∈ M : dist(x,B) ≤ ε}. Let X be a topological space, Y a metricspace, and F : X → Y a multivalued mapping with nonempty values. F is saidto be almost lower semicontinuous if given ε > 0, for each x ∈ X there is aneighborhood U(x) of x such that

⋂y∈U(x) Nε(F (y)) �= ∅. It is easy to check that

a mapping which is lower semicontinuous in the usual sense is also almost lowersemicontinuous.

Theorem 12 ([32]). Suppose X is a closed convex subset of a complete R-tree Y ,and suppose X is geodesically bounded. Let T : X → 2Y be an almost lowersemicontinuous mapping whose values are nonempty bounded closed convex subsetsof Y . Then there exists a point z ∈ X such that

dist(z, T (z)) = infx∈X

dist(x, T (z)).

The proof of Theorem 12 is based on Proposition 2 (see Section 2) and thefollowing selection theorem for R-trees.


Theorem 13 ([32]). Let X be a paracompact topological space, (Y, d) a complete R-tree, and T : X → 2Y an almost lower semicontinuous mapping whose values arenonempty bounded closed convex subsets of Y. Then T has a continuous selection.

5. Applications to graph theory

A graph is an ordered pair (V,E) where V is a set and E is a binary relationon V (E ⊆ V × V ). Elements of E are called edges. We are concerned here with(undirected) graphs that have a “loop” at every vertex (i.e., (a, a) ∈ E for eacha ∈ V ) and no “multiple” edges. Such graphs are called reflexive. In this caseE ⊆ V × V corresponds to a reflexive (and symmetric) binary relation on V.

For a graph G = (V,E) a map f : V → V is edge-preserving if (a, b) ∈ E⇒ (f(a), f(b)) ∈ E. For such a mapping we simply write f : G → G. Thereis a standard way of metrizing connected graphs: let each edge have length oneand take distance d(a, b) between two vertices a and b to be the length of theshortest path joining them. With this metric, edge-preserving mappings becomeprecisely the nonexpansive mappings. (Keep in mind that in a reflexive graph anedge-preserving map may collapse edges between distinct points since loops areallowed.)

We now turn to the classical Fixed Edge Theorem and show how it is aconsequence of results of the preceding section.

Theorem 14 ([34]). Let G be a reflexive graph that is connected, contains no cycles,and contains no infinite paths. Then every edge-preserving map of G into itself fixesan edge.

Proof. Suppose f : G → G is edge-preserving. Since a connected graph with nocycles is a tree, one can construct from the graph G an R-tree T by identifying each(nontrivial) edge with a unit interval of the real line and assigning the shortest pathdistance to any two points of T. It is easy to see that with this metric T is complete.One can now extend f affinely on each edge to the corresponding unit interval,and the resulting mapping f is a nonexpansive (hence continuous) mapping ofT → T. Thus f has a fixed point z by Theorem 7. Moreover, since T has uniquemetric segments and f is nonexpansive, the fixed point set F of f is convex (andclosed). It follows that either F contains a vertex of G, or z is the midpoint of aunit interval of T , in which case f must leave the corresponding edge fixed. �

An application of Baillon’s theorem about commuting families of nonexpan-sive mappings in hyperconvex metric space tells us even more. For details, see [17].

Theorem 15 ([17]). Let G be a reflexive graph that is connected, contains no cycles,and contains no infinite paths. Suppose F is a commuting family of edge-preservingmappings of G into itself. Then either:(a) there is a unique edge in G that is left fixed by each member of F; or(b) some vertex of G is left fixed by each member of F.


It is likely that the above result is known in a more abstract framework. Thisseems to be a natural formulation resulting from the metric context.

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W. A. KirkDepartment of MathematicsUniversity of IowaIowa City, IA 52242, USAe-mail: [email protected]

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Somerecentresultsinmetricﬁxedpointtheory · pair of vertices (the edges of ). A comparison triangle for the geodesic triangle (x 1,x 2,x 3)in(X,d) is a triangle (x 1,x 2,x 3):=