Geodesic rays, the "Lion-Man" game, and the fixed point property
aa r X i v : . [ m a t h . M G ] M a y Geodesic rays, the “Lion-Man” game, and the fixed point property
Genaro López-Acedo a , Adriana Nicolae b , Bożena Piątek c a Department of Mathematical Analysis - IMUS, University of Seville, Sevilla, Spain b Department of Mathematics, Babeş-Bolyai University, Kogălniceanu 1, 400084 Cluj-Napoca, Romania c Institute of Mathematics, Silesian University of Technology, 44-100 Gliwice, PolandE-mail addresses: [email protected] (G. López-Acedo), [email protected] (A. Nicolae), [email protected] (B. Piątek).
Abstract
This paper focuses on the relation among the existence of different types of curves (such as directionalones, quasi-geodesic or geodesic rays), the (approximate) fixed point property for nonexpansive mappings,and a discrete lion and man game. Our main result holds in the setting of
CAT(0) spaces that areadditionally Gromov hyperbolic.
Given a metric space X , the existence of either a geodesic ray in X or, more generally, of a curve thatapproximates a geodesic ray (e.g., a directional curve or a quasi-geodesic ray, see Section 3 for precisedefinitions), is connected to some intrinsic topological and geometric properties of the space. Such propertiesinclude the presence of upper or lower curvature bounds in the sense of Alexandrov [5, 12, 24], the Gromovhyperbolicity condition [5], the betwenness property [23], or, for normed spaces, the reflexivity [34]. Onthe other hand, the existence of different types of curves can be used in the analysis of several problemsamong which we mention the characterization of the fixed point property either for continuous [20, 19, 24]or nonexpansive mappings [34, 12, 27], and some pursuit-evasion games [2, 23].Geometric properties of a set stand behind the existence of fixed points for continuous or nonexpansivemappings defined on the set in question. This fact has prompted a very fruitful research direction because,among other reasons, it leads to challenging questions regarding the geometry of Banach spaces (see, e.g.,the monographs [3, 13]). Klee [20] was a pioneer in the use of topological rays as a tool to characterizecompactness of convex subsets of a locally convex metrizable linear topological space by means of the fixedpoint property for continuous mappings (see also [10]). Counterparts of this result for geodesic metricspaces were proved in [24, 25]. The notion of directional curves was introduced in [34] to characterize theapproximate fixed point property for nonexpansive mappings in convex subsets of a class of metric spaceswhich includes, in particular, Banach spaces or complete Busemann convex geodesic spaces. Further resultsin this direction were proved in [19, 27]. Namely, the absence of geodesic rays from a closed and convex set isequivalent to its fixed point property for continuous mappings in complete R -trees, while for the more generalsetting of complete CAT( κ ) spaces with κ < , it is equivalent to its fixed point property for nonexpansivemappings.Persuit-evasion problems are not only interesting because their analysis requires the use of tools fromdifferent areas of mathematics, but also because of their application in other disciplines such as robotics orthe modeling of animal behavior. In a recent survey, Chung et al. [8] classify persuit-evasion games based onthe environment where the game is played, the information available to the players, the restrictions imposedon the players’ motion, and the way of defining capture.The game that we consider in this work was proposed in [1, 2] (see also [4]) and is a discrete variant ofthe classical lion and man game which goes back to Rado (see [22]). This discrete equal-speed game withan ε -capture criterion, whose precise rules are described at the beginning of Section 4, will be called in thesequel the Lion-Man game. A first step in the study of the role that geodesic rays play in the analysis of theLion-Man game, and consequently its connection with the fixed point property, was given in [23], where it1as shown that in complete, locally compact, uniquely geodesic spaces, assuming a strongly convex domain,the success of the lion, the fixed point property for continuous mappings, and the compactness of the domainare all equivalent (see also [36]). Further results also from the quantitative point of view have been obtainedvery recently in [21]. In this work we analyze the relation among the existence of different types of curves anddeduce, in light of this analysis, connections among the solution of the Lion-Man game, the (approximate)fixed point property for nonexpansive mappings, and the existence of geodesic rays in the domain where thegame is played.The organization of the paper is as follows. After recalling in Section 2 some basic notions on geodesicmetric spaces, we study in Section 3 different types of curves which are related to the (approximate) fixedpoint property for nonexpansive mappings and to the Lion-Man game. More precisely, we focus on directionalcurves, local quasi-geodesic, quasi-geodesic and geodesic rays. Of particular importance for our main resultis Corollary 3.7 which states, in broad lines, that in the setting of complete Busemann convex spaces thatare additionally Gromov hyperbolic, the existence of a local quasi-geodesic ray implies the existence of ageodesic ray. Section 4 contains the main result of this work, Theorem 4.3, which shows that in the settingof complete CAT(0) spaces that are additionally Gromov hyperbolic, for a closed convex domain A , thefollowing are equivalent: A does not contain geodesic rays, A has the fixed point property for nonexpansivemappings, and the lion always wins the Lion-Man game played in A . We complete the section with somecomments and a result about the finite termination of the game in the setting of R -trees. We include here a brief despription of some of the notions and properties of geodesic metric spaces that wewill use in the following sections (for a detailed discussion, see, e.g., [5, 26]).Let ( X, d ) be a metric space. For x ∈ X and a nonempty subset A of X , we denote the distance of x to A by dist ( x, A ) = inf { d ( x, a ) : a ∈ A } and the metric projection of x onto A by P A ( x ) = { y ∈ A : d ( x, y ) = dist ( x, A ) } .Let x, y ∈ X . A geodesic joining x and y is a mapping γ : [ a, b ] ⊆ R → X such that γ ( a ) = x , γ ( b ) = y and d ( γ ( s ) , γ ( t )) = | s − t | for all s, t ∈ [ a, b ] . In this case we also say that γ starts from x . The image γ ([ a, b ]) of a geodesic γ joining x and y is called a geodesic segment joining x and y . A point z ∈ X belongsto a geodesic segment joining x and y if and only if there exists t ∈ [0 , such that d ( z, x ) = td ( x, y ) and d ( z, y ) = (1 − t ) d ( x, y ) . In this case z = γ ((1 − t ) a + tb ) , where γ : [ a, b ] → X is a geodesic joining x and y (that starts from x ) and whose image is the geodesic segment in question. We denote a geodesic segmentjoining x and y by [ x, y ] . Note however that, in general, geodesic segments between two given points mightnot be unique. If every two points in X are joined by a (unique) geodesic segment, we say that X is a (uniquely) geodesic space . A subset A of a geodesic space is convex if given two points in A , every geodesicsegment joining them is contained in A .If in the definition of a geodesic, instead of the interval [ a, b ] , one considers [0 , ∞ ) , then the image of γ is called a geodesic ray with the remark that sometimes we also refer to the mapping γ itself as a geodesicray. We say that a subset of a geodesic space is geodesically bounded if it does not contain any geodesic ray.In normed spaces, algebraic segments are geodesic segments and half-lines are geodesic rays. Thus, everynormed space is a geodesic space. Moreover, a normed space is uniquely geodesic if and only if it is strictlyconvex. In this case, the geodesic segments coincide with the algebraic segments, while the geodesic rays areprecisely the half-lines.A convex subset of a normed space E is called linearly bounded if it has a bounded intersection with allthe lines in E (see, e.g., [33]). Note that in strictly convex normed spaces, the notions of linear boundednessand geodesic boundedness agree.Suppose next that ( X, d ) is a geodesic space. We say that X is Busemann convex if given any twogeodesics γ : [ a, b ] → X and σ : [ c, d ] → X , d ( γ ((1 − t ) a + tb ) , σ ((1 − t ) c + td )) ≤ (1 − t ) d ( γ ( a ) , σ ( c )) + td ( γ ( b ) , σ ( d )) for any t ∈ [0 , . Every Busemann convex space is uniquely geodesic. In addition, a normed space is Busemann convex if andonly if it is strictly convex. 2or κ ∈ R , let M κ be the complete, simply connected -dimensional Riemannian manifold of constantsectional curvature κ . In the sequel we assume that κ ≤ .A geodesic triangle ∆ = ∆( x , x , x ) in X consists of three points x , x , x ∈ X (its vertices ) andthree geodesic segments (its sides ) joining each pair of points. A comparison triangle for ∆ is a triangle ∆ = ∆( x , x , x ) in M κ with d ( x i , x j ) = d M κ ( x i , x j ) for i, j ∈ { , , } . For κ fixed, comparison triangles ofgeodesic triangles always exist and are unique up to isometry.Let γ : [ a, b ] → X and σ : [ c, d ] → X be two nonconstant geodesics that start from the same point x = γ ( a ) = σ ( c ) . For t ∈ ( a, b ] , s ∈ ( c, d ] , and a geodesic triangle ∆( x, γ ( t ) , σ ( s )) , consider a comparisontriangle ∆( x, γ ( t ) , σ ( s )) in R = M and denote its interior angle at x by ∠ x ( γ ( t ) , σ ( s )) . The Alexandrovangle ∠ ( γ, σ ) between the geodesics γ and σ is defined as ∠ ( γ, σ ) = lim sup t,s → ∠ x ( γ ( t ) , σ ( s )) . For x, y, z ∈ X with x = y and x = z , if both the points x and y , and x and z , are joined by a uniquegeodesic segment, then we also denote the corresponding Alexandrov angle by ∠ x ( y, z ) .If γ , γ , γ are three geodesics that start from the same point, then ∠ ( γ , γ ) ≤ ∠ ( γ , γ ) + ∠ ( γ , γ ) . In particular, let γ : [ a, b ] → X be a nonconstant geodesic and c ∈ ( a, b ) . Define γ : [ a, c ] → X by γ ( t ) = γ ( a + c − t ) and γ : [ c, b ] → X by γ ( t ) = γ ( t ) . If γ is a nonconstant geodesic that starts from γ ( c ) ,then ∠ ( γ , γ ) + ∠ ( γ , γ ) ≥ π .A geodesic triangle ∆ in X satisfies the CAT( κ ) inequality if for every comparison triangle ∆ in M κ of ∆ and for every x, y ∈ ∆ we have d ( x, y ) ≤ d M κ ( x, y ) , where x, y ∈ ∆ are the comparison points of x and y , i.e., if x belongs to the side joining x i and x j , then x belongs to the side joining x i and x j such that d ( x i , x ) = d M κ ( x i , x ) .A CAT( κ ) space is geodesic space where every geodesic triangle satisfies the CAT ( κ ) inequality. CAT( κ ) spaces are also known as spaces of curvature bounded above by κ (in the sense of Alexandrov). In any CAT( κ ) space there exists a unique geodesic joining each pair of points.An R -tree is a uniquely geodesic space X such that if x, y, z ∈ X with [ y, x ] ∩ [ x, z ] = { x } , then [ y, x ] ∪ [ x, z ] = [ y, z ] . It is easily seen that a metric space is an R -tree if and only if it is a CAT( κ ) space for anyreal κ ≤ .Another geometric condition that plays an essential role in the sequel is that of δ -hyperbolicity. Thereare several ways to introduce this concept and we follow here the one attributed to Rips (see [5, p. 399]).Given M ≥ , a geodesic triangle in a metric space is called M -slim if any of its sides is contained in the M -neighborhood of the union of the other two sides. A geodesic space X is called δ -hyperbolic for some δ ≥ if every geodesic triangle in it is δ -slim. If a geodesic space is δ -hyperbolic for some δ ≥ , then it isalso said to be Gromov hyperbolic . CAT( κ ) spaces with κ < are δ -hyperbolic , where δ only depends on κ .Moreover, a geodesic space is an R -tree if and only if it is -hyperbolic. Note also that there exist CAT(0) spaces which are not δ -hyperbolic such as Hilbert spaces.Given three points x, y, z in a metric space, the Gromov product ( y | z ) x is the nonnegative number definedby ( y | z ) x = 12 ( d ( x, y ) + d ( x, z ) − d ( y, z )) . The following characterization of δ -hyperbolicity is due Gromov and often used as an alternative definition(see [7, Lemma 1.2.3 and Exercise 1.2.4]). Proposition 2.1.
A geodesic space ( X, d ) is δ -hyperbolic for some δ ≥ if and only if there exists δ ′ ≥ such that for all three points x, y, z ∈ X , fixing any geodesic segments [ x, y ] and [ x, z ] joining x and y , and x and z , respectively, the following implication holds: if y ′ ∈ [ x, y ] and z ′ ∈ [ x, z ] are such that d ( x, y ′ ) = d ( x, z ′ ) ≤ ( y | z ) x , then d ( y ′ , z ′ ) ≤ δ ′ . Geodesic rays and the fixed point property
Let ( X, d ) be a metric space and A ⊆ X . A mapping T : A → X is called nonexpansive if d ( T x, T y ) ≤ d ( x, y ) for all x, y ∈ A . We say that A has the fixed point property (FPP for short) if each nonexpansive mapping T : A → A has at least one fixed point, i.e., a point x ∈ A such that T x = x . A very well-known resultfrom 1965 proved independently by Browder [6], Göhde [15] and Kirk [16] says that every closed, convexand bounded subset of a Hilbert space has the FPP. In 1980, Ray [30] approached the converse problem andproved that boundedness is a necessary condition for a closed and convex subset of a Hilbert space to havethe FPP. Similar results in the setting of Banach spaces can be found, e.g., in [9, 31, 32, 35].After the publication of the papers [17, 18] due to Kirk, geodesic metric spaces have called the attentionof many authors working in metric fixed point theory. Especially relevant in the study of the FPP provedto be the existence of upper bounds on the curvature in the sense of Alexandrov. Since Hilbert spaces arethe only Banach spaces which are CAT(0) , it was natural to consider the question whether the Browder-Göhde-Kirk theorem and Ray’s result mentioned before hold true in complete
CAT(0) spaces. Regarding theBrowder-Göhde-Kirk theorem, the answer is positive (see [17]), however Ray’s result fails as there exist broadclasses of
CAT(0) spaces where a closed and convex set has the FPP if and only if it is geodesically bounded(and hence not necessarily bounded). Such examples include the complex Hilbert ball with the hyperbolicmetric (see [14, Theorems 25.2, 32.2] and [27, Corollary 4.4]), complete R -trees (see [11, Theorem 4.3]),or even complete CAT( κ ) spaces with κ < (see [27, Corollary 4.2]). In fact, this characterization of theFPP in terms of geodesic boundedness holds in the setting of complete CAT(0) spaces that are additionally δ -hyperbolic (see [29, Corollary 3.2]). Other results related to the FPP of unbounded sets in geodesic spacescan be found in [12, 28, 27].Another more general property which was considered in this line is defined as follows: we say that A hasthe approximate fixed point property (AFPP for short) if inf { d ( x, T x ) : x ∈ A } = 0 for every nonexpansivemapping T : A → A . It is immediate that every closed, convex and bounded subset of a Banach space hasthe AFPP (see [13, Lemma 3.1]). However, there exist unbounded, closed and convex sets that have theAFPP. Reich [33] showed that a closed and convex subset of a reflexive Banach space has the AFPP if andonly if it is linearly bounded. Shafrir [34] used the notion of directional curve to characterize the AFPP ofconvex sets in a class of metric spaces which includes, in particular, Banach spaces or complete Busemannconvex geodesic spaces. Definition 3.1.
Let ( X, d ) be a metric space. A curve γ : [0 , ∞ ) → X is said to be directional if there exists b ≥ such that | s − t | − b ≤ d ( γ ( s ) , γ ( t )) ≤ | s − t | , for all s, t ≥ .A subset of X is called directionally bounded if it contains no directional curves.A sequence ( x n ) in X is said to be directional if the following two conditions hold:1. lim n →∞ d ( x , x n ) = ∞ ;2. there exists b ≥ such that d ( x n , x n l ) ≥ l − X i =1 d ( x n i , x n i +1 ) − b, (1)for all n < n < · · · < n l .Clearly, every geodesic ray is a directional curve, so directionally bounded sets are always geodesicallybounded. Note also that a convex subset of a geodesic space is directionally bounded if and only if it doesnot contain any directional sequence (see [34, Lemma 2.3]).In Banach spaces or in complete Busemann convex spaces, a convex set has the AFPP if and only if itis directionally bounded (see [34, Theorem 2.4]). Moreover, the directional boundedness can also be usedto give a characterization of reflexivity in Banach spaces. Namely, a Banach space is reflexive if and only ifevery closed and convex subset of it that is linearly bounded is directionally bounded (see [34, Proposition3.5]). In the nonlinear case we have the following result (a corresponding one for the case of a Busemannconvex space that is additionally δ -hyperbolic is given in Proposition 3.8).4 roposition 3.2. If ( X, d ) is a complete CAT(0) space, then every closed and convex subset of X that isgeodesically bounded is directionally bounded.Proof. Let A be a closed and convex subset of X . We show that if A is not directionally bounded, thenit is not geodesically bounded either. Take ( x n ) a directional sequence in A with constant b . For n ∈ N ,denote d n = d ( x , x n ) . Then lim n →∞ d n = ∞ and, by (1), for m, n ∈ N with < m < n we have d n ≥ d m + d ( x m , x n ) − b .Let m, n ∈ N with < m < n such that d m , d n > . For the geodesic triangle ∆( x , x m , x n ) , considera comparison triangle ∆( x , x m , x n ) in R and denote its interior angle at x by α m,n . Let b m,n = d m + d ( x m , x n ) − d n ∈ [0 , b ] . The cosine law in R yields b m,n + d m + d n − d m b m,n − d m d n + 2 d n b m,n = ( b m,n − d m + d n ) = d ( x m , x n ) = d m + d n − d m d n cos α m,n , from where b + 2 d n b ≥ b m,n + 2 d n b m,n ≥ b m,n + 2 d n b m,n − d m b m,n = 2 d m d n (1 − cos α m,n ) , so sin α m,n ≤ b d m (cid:18) b d n + 1 (cid:19) . For n, k ≥ , if d n ≥ k , pick x nk ∈ [ x , x n ] so that d ( x , x nk ) = k . Let k ≥ and ε > . Then there exists n k ≥ such that for all n, m ≥ n k , d m , d n ≥ k and α m,n < ε . If n, m ≥ n k , denote x mk and x nk the comparisonpoints in ∆( x , x m , x n ) of x mk and x nk , respectively. As ( x nk ) n ≥ n k is Cauchy and d ( x mk , x nk ) ≤ d ( x mk , x nk ) forall n, m ≥ n k , it follows that ( x nk ) n ≥ n k is Cauchy too, hence it converges to some x ∗ k ∈ A .Let k, l ≥ with k < l . For n sufficiently large, x nk = (1 − k/l ) x + ( k/l ) x nl . By Busemann convexity, d ( x nk , (1 − k/l ) x + ( k/l ) x ∗ l ) ≤ kl d ( x nl , x ∗ l ) , which shows that x ∗ k = (1 − k/l ) x + ( k/l ) x ∗ l , so x ∗ k ∈ [ x , x ∗ l ] . Finally, S k [ x , x ∗ k ] ⊆ A is a geodesic ray.The following notions will play an important role in the proof of our main result. Definition 3.3.
Let ( X, d ) be a metric space, λ ≥ , ε ≥ , and k > . A ( λ, ε ) -quasi-geodesic is a mapping γ : [ a, b ] ⊆ R → X such that λ | s − t | − ε ≤ d ( γ ( s ) , γ ( t )) ≤ λ | s − t | + ε, (2)for all s, t ∈ [ a, b ] . We say that γ joins γ ( a ) and γ ( b ) . If instead of the interval [ a, b ] one considers [0 , ∞ ) ,then γ (or its image) is called a ( λ, ε ) -quasi-geodesic ray . If γ is a ( λ, ε ) -quasi-geodesic for every ε > , thenwe simply say that γ is a λ -quasi-geodesic . At the same time, a λ -quasi-geodesic defined on [0 , ∞ ) is calleda λ -quasi-geodesic ray .A k -local ( λ, ε ) -quasi-geodesic is a mapping γ : [ a, b ] ⊆ R → X such that (2) holds for all s, t ∈ [ a, b ] with | s − t | ≤ k . The “ k -local” versions of the other notions from above are defined in a similar way.It is clear that every geodesic is a λ -quasi-geodesic for any λ ≥ . Furthermore, directional curves withconstant b are (1 , b ) -quasi-geodesic rays. However, not all quasi-geodesic rays are directional curves (to seethis, one can consider the curve given in [5, p. 142, Exercise 8.23]).For our main result, the question whether a local quasi-geodesic ray is actually, up to an appropriatechange of constants, a quasi-geodesic ray is particularly relevant. Although a positive answer can be antic-ipated from [5, p. 407, Remark], for completeness we clarify this aspect in Proposition 3.5 where we use aproof strategy similar to the one of [5, p. 405, Theorem 1.13].Before stating this result, we recall that given a metric space ( X, d ) , a ( λ, ε ) -quasi-geodesic triangle ∆ = ∆( x , x , x ) in X , where λ ≥ and ε ≥ , consists of three points x , x , x ∈ X (its vertices ) and theimages of three ( λ, ε ) -quasi-geodesics (its sides ) joining each pair of points. A ( λ, ε ) -quasi-geodesic triangleis called M -slim , where M ≥ , if each of its sides is contained in the M -neighborhood of the union of theother two sides. As before, one can also consider the notion of λ -quasi-geodesic triangle and these are in factthe triangles that we will work with. The next property follows from [5, p. 402, Corollary 1.8].5 emark . If X is a δ -hyperbolic geodesic space, then for every λ ≥ there exists M = M ( δ, λ ) such thatall λ -quasi-geodesic triangles are M -slim. Proposition 3.5.
Let ( X, d ) be a δ -hyperbolic geodesic space, λ ≥ , and M = M ( δ, λ ) be given by Remark3.4. If γ is a k -local λ -quasi-geodesic ray such that k > λM , then γ is a ( λ ∗ , ε ) -quasi-geodesic ray, where λ ∗ = (cid:18) λ − Mk/ λM (cid:19) − and ε = 2 M. Proof.
One can easily see that λ ∗ ≥ . To prove the result, it is enough to show that for every a, b ∈ [0 , ∞ ) , γ | [ a,b ] is a ( λ ∗ , ε ) -quasi-geodesic. Moreover, we can assume that b − a > k (otherwise the conclusion isimmediate because λ ≤ λ ∗ ). Fix a geodesic segment joining γ ( a ) and γ ( b ) and denote it by [ γ ( a ) , γ ( b )] . Claim 1.
The image of γ | [ a,b ] is contained in a M -neighborhood of [ γ ( a ) , γ ( b )] .Proof of Claim 1. Let x = γ ( t ) , where t ∈ [ a, b ] , such that dist( x, [ γ ( a ) , γ ( b )]) = max s ∈ [ a,b ] dist( γ ( s ) , [ γ ( a ) , γ ( b )]) . We know that max { t − a, b − t } > λM (since b − a > k > λM ) . Depending on the values of t − a and t − b , we distinguish two situations.Case I : t − a, b − t > λM .Take t , t ∈ [ a, b ] such that t − t , t − t ∈ (4 λM, k/ . Note that t ∈ ( t , t ) . Furthermore, t − t < k and so γ | [ t ,t ] is a λ -quasi geodesic. Denote y = γ ( t ) , z = γ ( t ) , y ′ ∈ P [ γ ( a ) ,γ ( b )] ( y ) , and z ′ ∈ P [ γ ( a ) ,γ ( b )] ( z ) .Then d ( x, y ) = d ( γ ( t ) , γ ( t )) ≥ λ ( t − t ) > M. (3)Similarly, d ( x, z ) > M . Consider two geodesic segments [ y, y ′ ] and [ y ′ , z ] joining y and y ′ , and y ′ and z ,respectively. Form the λ -quasi-geodesic triangle ∆( y ′ , y, z ) whose sides are [ y, y ′ ] , [ y ′ , z ] and the image of the λ -quasi-geodesic γ | [ t ,t ] . Then ∆( y ′ , y, z ) is M -slim, so there exists u ∈ [ y, y ′ ] ∪ [ y ′ , z ] such that d ( x, u ) ≤ M .Suppose u ∈ [ y ′ , z ] . Take a geodesic segment [ z, z ′ ] joining z and z ′ and the geodesic segment [ y ′ , z ′ ] ⊆ [ γ ( a ) , γ ( b )] joining y ′ and z ′ . The geodesic triangle ∆( y ′ , z, z ′ ) whose sides are [ y ′ , z ] , [ z, z ′ ] and [ y ′ , z ′ ] is M -slim, so there exists w ∈ [ z, z ′ ] ∪ [ y ′ , z ′ ] such that d ( u, w ) ≤ M .Thus, there exists p ∈ [ y, y ′ ] ∪ [ z, z ′ ] ∪ [ y ′ , z ′ ] such that d ( x, p ) ≤ M. (4)Now we show that actually p ∈ [ y ′ , z ′ ] . If p ∈ [ y, y ′ ] , we have d ( x, y ′ ) ≤ d ( x, p ) + d ( p, y ′ ) = d ( x, p ) + d ( y, y ′ ) − d ( y, p ) ≤ d ( x, p ) + d ( y, y ′ ) − ( d ( x, y ) − d ( x, p ))= 2 d ( x, p ) + d ( y, y ′ ) − d ( x, y ) < d ( y, y ′ ) by (4) and (3) . This contradicts the choice of x . In a similar way one obtains again a contradiction if p ∈ [ z, z ′ ] .Case II: t − a > λM and b − t ≤ λM . (The case b − t > λM and t − a ≤ λM is dealt with in asimilar way.)Take t ∈ [ a, b ] such that t − t ∈ (4 λM, k/ . Note that t ∈ ( t , b ] . Furthermore, b − t = b − t + t − t < λM + k < k, so γ | [ t ,b ] is a λ -quasi-geodesic. Denote y = γ ( t ) and y ′ ∈ P [ γ ( a ) ,γ ( b )] ( y ) . Then d ( x, y ) = d ( γ ( t ) , γ ( t )) ≥ λ | t − t | > M. [ y, y ′ ] joining y and y ′ and the geodesic segment [ y ′ , γ ( b )] ⊆ [ γ ( a ) , γ ( b )] joining y ′ and γ ( b ) . Form the λ -quasi-geodesic triangle ∆( y ′ , y, γ ( b )) whose sides are [ y, y ′ ] , [ y ′ , γ ( b )] and theimage of the λ -quasi-geodesic γ | [ t ,b ] . This triangle is M -slim, so there exists p ∈ [ y, y ′ ] ∪ [ y ′ , γ ( b )] such that d ( x, p ) ≤ M . In fact, p ∈ [ y ′ , γ ( b )] because if p ∈ [ y, y ′ ] we get a contradiction as in the previous case.Hence, in both cases, we find p ∈ [ γ ( a ) , γ ( b )] such that d ( x, p ) ≤ M . This finishes the proof of theclaim.Now we show that γ | [ a,b ] is a ( λ ∗ , ε ) -quasi-geodesic. Dividing the k -local λ -quasi-geodesic γ | [ a,b ] intosufficiently small subpaths and using the triangle inequality we get d ( γ ( s ) , γ ( t )) ≤ λ | s − t | ≤ λ ∗ | s − t | , for all s, t ∈ [ a, b ] . Thus, we only need to prove the left-hand inequality in the definition of a ( λ ∗ , ε ) -quasi-geodesic. Bearing in mind that a subpath of a k -local λ -quasi-geodesic is a k -local λ -quasi-geodesic as well,it is enough to prove that λ ∗ ( b − a ) − ε ≤ d ( γ ( a ) , γ ( b )) . Let n = ⌊ ( b − a ) / ( k/ λM ) ⌋ . Note that n ≥ as b − a > k > k/ λM . Take t = a and for i ∈ { , . . . , n − } , let t i +1 = t i + k λM ∈ [ a, b ] . Then γ | [ t i ,t i +1 ] is a λ -quasi-geodesic for all i ∈ { , . . . , n − } . Likewise, γ | [ t n ,b ] is a λ -quasi-geodesic. Claim 2.
Suppose n ≥ , fix i ∈ { , . . . , n − } , denote x = γ ( t i ) , m = γ ( t i +1 ) , y = γ ( t i +2 ) , and take thecorresponding projections onto [ γ ( a ) , γ ( b )] x ′ ∈ P [ γ ( a ) ,γ ( b )] ( x ) , m ′ ∈ P [ γ ( a ) ,γ ( b )] ( m ) , y ′ ∈ P [ γ ( a ) ,γ ( b )] ( y ) . Then m ′ ∈ [ x ′ , y ′ ] ⊆ [ γ ( a ) , γ ( b )] .Proof of Claim 2. By Claim 1, d ( x, x ′ ) ≤ M . Additionally, d ( x, m ) ≥ λ (cid:18) k λM (cid:19) > M. Hence, M < d ( x, m ) ≤ d ( x, x ′ ) + d ( x ′ , m ) ≤ M + d ( x ′ , m ) , so d ( x ′ , m ) > M . Similarly, d ( y ′ , m ) > M .Denote x = γ ( t i + λM ) and y = γ ( t i +1 + k/ . Take four geodesic segments [ x ′ , x ] , [ x ′ , x ] , [ x ′ , m ] and [ x ′ , y ] that join x ′ and the respective points x , x , m and y . Let also [ y ′ , y ] be a geodesic segmentjoining y ′ and y . Consider the λ -quasi-geodesic triangle ∆( x ′ , x , y ) of sides [ x ′ , x ] , [ x ′ , y ] and the imageof γ | [ t i + λM,t i +1 + k/ . Let also ∆( x ′ , y , y ′ ) be the geodesic triangle of sides [ x ′ , y ] , [ y ′ , y ] and [ x ′ , y ′ ] . Since ∆( x ′ , x , y ) is M -slim, there exists u ∈ [ x ′ , x ] ∪ [ x ′ , y ] such that d ( m, u ) ≤ M . If u ∈ [ x ′ , y ] , then,as ∆( x ′ , y , y ′ ) is M -slim, there exists w ∈ [ y ′ , y ] ∪ [ x ′ , y ′ ] such that d ( u, w ) ≤ M . Hence, one can find p ∈ [ x ′ , x ] ∪ [ y ′ , y ] ∪ [ x ′ , y ′ ] so that d ( m, p ) ≤ M .Suppose p ∈ [ x ′ , x ] . Consider the M -slim λ -quasi-geodesic triangle ∆( x ′ , x , x ) of sides [ x ′ , x ] , [ x ′ , x ] andthe image of γ | [ t i ,t i + λM ] . If d ( p, u ) ≤ M for some u ∈ [ x ′ , x ] , then d ( p, x ) ≤ d ( p, u )+ d ( u, x ) ≤ M +2 M = 3 M .But then M < d ( m, x ) ≤ d ( m, p ) + d ( p, x ) ≤ M + 3 M = 5 M, a contradiction. If d ( p, u ) ≤ M for some u = γ ( t i + α ) , where α ∈ [0 , λM ] , then d ( u, m ) = d ( γ ( t i + α ) , γ ( t i + k/ λM ) ≥ λ (cid:18) k λM − α (cid:19) ≥ k λ > M. d ( u, m ) ≤ d ( u, p ) + d ( p, m ) ≤ M + 2 M = 3 M , which is again a contradiction. This shows that p / ∈ [ x ′ , x ] . Similarly, p / ∈ [ y ′ , y ] , so p ∈ [ x ′ , y ′ ] .Let [ m, m ′ ] and [ m, p ] be two geodesic segments joining m and m ′ , and m and p , respectively. Thegeodesic triangle ∆( m, m ′ , p ) whose sides are [ m, m ′ ] , [ m, p ] and [ m ′ , p ] ⊆ [ γ ( a ) , γ ( b )] is M -slim. Recall that,by Claim 1, d ( m, m ′ ) ≤ M . Thus, d ( u, m ) ≤ M for all u ∈ [ m ′ , p ] . This shows that x ′ , y ′ / ∈ [ m ′ , p ] and, as p ∈ [ x ′ , y ′ ] , we must have m ′ ∈ [ x ′ , y ′ ] .Denote now p i ∈ P [ γ ( a ) ,γ ( b )] γ ( t i ) for i ∈ { , · · · , n } . Then (cid:18) k λM (cid:19) λ ≤ d ( γ ( t i ) , γ ( t i +1 )) ≤ d ( γ ( t i ) , p i ) + d ( p i , p i +1 ) + d ( p i +1 , γ ( t i +1 )) ≤ M + d ( p i , p i +1 ) , and hence, d ( p i , p i +1 ) ≥ k λ − M, for all i ∈ { , · · · , n − } . Likewise, d ( p n , γ ( b )) ≥ d ( γ ( t n ) , γ ( b )) − d ( p n , γ ( t n )) ≥ λ ( b − t n ) − M. Therefore, using Claim 2, d ( γ ( a ) , γ ( b )) = n − X i =0 d ( p i , p i +1 ) + d ( p n , γ ( b )) ≥ n (cid:18) k λ − M (cid:19) + 1 λ ( b − t n ) − M = 1 λ (cid:18) n k b − t n (cid:19) − nM − M = b − aλ − nM − M ≥ b − aλ − M b − ak/ λM − M = (cid:18) λ − Mk/ λM (cid:19) ( b − a ) − M = 1 λ ∗ ( b − a ) − ε. We see next that if the δ -hyperbolic geodesic space is also Busemann convex, then the existence of aquasi-geodesic ray yields the existence of a geodesic ray. Combined with Proposition 3.5, this shows that forthe existence of a geodesic ray, the presence a local quasi-geodesic ray is sufficient. Proposition 3.6.
Let ( X, d ) be a complete Busemann convex space that is additionally δ -hyperbolic and A ⊆ X closed and convex. If A contains a ( λ, ε ) -quasi-geodesic ray, where λ ≥ and ε ≥ , then it containsa geodesic ray.Proof. Let γ : [0 , ∞ ) → A be a ( λ, ε ) -quasi-geodesic ray. Then λ | s − t | − ε ≤ d ( γ ( s ) , γ ( t )) ≤ λ | s − t | + ε, for all s, t ∈ [0 , ∞ ) .Take α > such that β = 1 λ + λ + α (cid:18) λ − λ (cid:19) > . Let x = γ (0) and for n ≥ , define x n = γ ( α n ) . Then x n | x n +1 ) x ≥ λ ( α n + α n +1 ) − ε − λ ( α n +1 − α n ) − ε = βα n − ε. Thus, for n large enough, ( x n | x n +1 ) x ≥ βα n / . Observe also that lim n →∞ d ( x , x n ) = ∞ .For n, k ≥ , if d ( x , x n ) ≥ k , pick x nk ∈ [ x , x n ] so that d ( x , x nk ) = k . Fix k ≥ . For n sufficiently large, d ( x , x n ) , d ( x , x n +1 ) ≥ k and ( x n | x n +1 ) x ≥ k . Moreover, as ( x n | x n +1 ) x ≤ min { d ( x , x n ) , d ( x , x n +1 ) } , we8an take y n ∈ [ x , x n ] and y n +1 ∈ [ x , x n +1 ] such that d ( x , y n ) = d ( x , y n +1 ) = ( x n | x n +1 ) x . By Busemannconvexity, d ( x nk , x n +1 k ) ≤ k ( x n | x n +1 ) x d ( y n , y n +1 ) . Applying Proposition 2.1, we conclude that there exists δ ∗ > such that d ( x nk , x n +1 k ) ≤ kδ ∗ ( x n | x n +1 ) x ≤ kδ ∗ βα n , for all n sufficiently large. So the sequence ( x nk ) n is Cauchy and hence converges to a point x ∗ k ∈ A whichsatisfies d ( x , x ∗ k ) = k .Let k, l ≥ with k < l . As in the proof of Proposition 3.2 one can use Busemann convexity to show that x ∗ k ∈ [ x , x ∗ l ] , hence S k [ x , x ∗ k ] is a geodesic ray in A . Corollary 3.7.
Let ( X, d ) be a complete Busemann convex space that is additionally δ -hyperbolic and A ⊆ X closed and convex. Suppose λ ≥ and M = M ( δ, λ ) is given by Remark 3.4. If A contains a k -local λ -quasi-geodesic ray such that k > λM , then it contains a geodesic ray. Since any directional curve is a quasi-geodesic ray, we can apply Proposition 3.6 to get the followinganalogue of Proposition 3.2.
Proposition 3.8.
In a complete Busemann convex space that is additionally δ -hyperbolic, every closed andconvex set that is geodesically bounded is directionally bounded.Remark . Let us notice that the previous result is in general not true if the space is merely assumed to beBusemann convex. Actually, it is well-known that every separable Banach space has an equivalent strictlyconvex norm (see, e.g., [3, p. 60, Theorem 1.5]). In particular, one can renorm ℓ to make it Busemannconvex. Since this space is not reflexive, according to [34, Proposition 3.5], geodesic boundedness cannotimply directional boundedness for every closed and convex set.In contrast to δ -hyperbolic Busemann convex spaces, in Hilbert spaces, the existence of a quasi-geodesicray in a closed and convex set does not yield the existence of a geodesic ray as the following example shows.(Recall however that, by [34, Proposition 3.5], in any Hilbert space, the existence of a directional curveimplies the existence of a geodesic ray.) Example . Let A ⊆ ℓ be given by A = { x = ( x , x , x , . . . ) ∈ ℓ : 0 ≤ x n ≤ n for all n ≥ } . Then A is closed, convex and linearly bounded, hence geodesically bounded. We construct next a quasi-geodesic ray in A . To this end, consider first the sequence ( x k ) in A , where x = (0 , , , . . . ) and x kn = (cid:26) n if n ≤ k otherwise , for all k ≥ . Let a = 0 and a k = P kn =1 n for k ≥ . Note that a k +1 − a k = 10 k +1 = (cid:13)(cid:13) x k +1 − x k (cid:13)(cid:13) forall k ∈ N . Define now γ : [0 , ∞ ) → A , γ ( t ) = (cid:18) − t − a k k +1 (cid:19) x k + t − a k k +1 x k +1 , for a k ≤ t < a k +1 , where k ∈ N . Note that γ ( a k ) = x k for all k ∈ N . We show that γ is a p / -quasi-geodesic ray.Case I: Let k ≥ and a k − ≤ s ≤ t < a k . Then k γ ( t ) − γ ( s ) k = t − s .Case II: Let k ≥ , a k − ≤ s < a k , and a k ≤ t < a k +1 . Denote u = (cid:13)(cid:13) x k − γ ( s ) (cid:13)(cid:13) = a k − s and v = (cid:13)(cid:13) γ ( t ) − x k (cid:13)(cid:13) = t − a k . Then t − s = u + v ≤ √ p u + v = √ k γ ( t ) − γ ( s ) k k γ ( t ) − γ ( s ) k ≤ (cid:13)(cid:13) γ ( t ) − x k (cid:13)(cid:13) + (cid:13)(cid:13) x k − γ ( s ) (cid:13)(cid:13) = v + u = t − s. Case III: Let ≤ k < l , a k − ≤ s < a k , and a l ≤ t < a l +1 . Denote u = (cid:13)(cid:13) x k − γ ( s ) (cid:13)(cid:13) = a k − s, v = (cid:13)(cid:13) γ ( t ) − x l (cid:13)(cid:13) = t − a l , and w = (cid:13)(cid:13) x l − x k (cid:13)(cid:13) = vuut l − X i = k i +1) = 13 √ p l +1) − k +1) ≥ √ (cid:0) l +1 − k +1 (cid:1) = 3 √ l − X i = k i +1 = 3 √ l − X i = k (cid:13)(cid:13) x i +1 − x i (cid:13)(cid:13) . Then t − s = a k − s + l − X i = k ( a i +1 − a i ) + t − a l = u + v + l − X i = k (cid:13)(cid:13) x i +1 − x i (cid:13)(cid:13) ≤ u + v + √ w ≤ √
113 ( u + v + w ) ≤ r p u + v + w = r k γ ( t ) − γ ( s ) k , where the last inequality follows from u + v + w ) ≥ ( u + v + w ) .At the same time, k γ ( t ) − γ ( s ) k ≤ (cid:13)(cid:13) γ ( t ) − x l (cid:13)(cid:13) + (cid:13)(cid:13) x k − γ ( s ) (cid:13)(cid:13) + l − X i = k (cid:13)(cid:13) x i +1 − x i (cid:13)(cid:13) = t − a l + a k − s + l − X i = k ( a i +1 − a i ) = t − s. Let ( X, d ) be a uniquely geodesic space and A ⊆ X nonempty and convex. Take D > and suppose that L , M ∈ A are the starting points of the lion and the man, respectively. At step n + 1 , n ∈ N , the lion movesfrom the point L n to the point L n +1 ∈ [ L n , M n ] such that d ( L n , L n +1 ) = min { D, d ( L n , M n ) } . The manmoves from the point M n to the point M n +1 ∈ A satisfying d ( M n , M n +1 ) ≤ D . We say that the lion wins ifthe sequence ( d ( L n +1 , M n )) converges to . Otherwise the man wins. Denote in the sequel D n = d ( L n , M n ) , n ∈ N .It is easy to see that the lion wins if and only if either of the following two mutually exclusive situationsholds:(1) there exists n ∈ N such that D n ≤ D . In this case, L n +1 = M n for all n ≥ n ;(2) D n > D for all n ∈ N and lim n →∞ D n = D . Note that the last limit exists because in this case thesequence ( D n ) is nonincreasing as D n +1 ≤ d ( L n +1 , M n ) + d ( M n , M n +1 ) = D n − D + d ( M n , M n +1 ) ≤ D n , for all n ∈ N .Consequently, the man wins if and only if D n > D for all n ∈ N and lim n →∞ D n > D . Theorem 4.1.
Let ( X, d ) be a uniquely geodesic space and A ⊆ X a nonempty and convex set where theLion-Man game is played following the rules described above. If the lion always wins, then A is directionallybounded. roof. Assume that there exists a directional curve γ : [0 , ∞ ) → A , i.e., there exists b > such that | s − t | − b ≤ d ( γ ( s ) , γ ( t )) ≤ | s − t | , for all s, t ≥ . Take D = b , L = γ (0) and M n = γ (( n + 2) D + 1) for all n ≥ . Then, for every n ≥ , d ( M n , M n +1 ) ≤ ( n + 3) D + 1 − ( n + 2) D − D,d ( L , L n ) ≤ n − X i =0 d ( L i , L i +1 ) ≤ nD and d ( M n , L n ) ≥ d ( M n , L ) − d ( L , L n ) ≥ ( n + 2) D + 1 − D − nD ≥ D + 1 , hence the man wins.The subsequent result shows that if the Lion-Man game is played in a CAT(0) space, then the success ofthe man yields the existence of a local quasi-geodesic ray.
Proposition 4.2.
Let ( X, d ) be a CAT(0) space and A ⊆ X a nonempty and convex set where the Lion-Mangame is played. If the man wins, then for every k > there exists a k -local √ -quasi-geodesic ray in A .Proof. Suppose the man wins. For n ∈ N with n ≥ , denote β n = ∠ L n ( L n − , L n +1 ) = ∠ L n ( L n − , M n ) .The following claim is also justified in [2, p. 281]. Claim 1. lim n →∞ β n = π .Proof of Claim 1. Let α n = ∠ L n ( M n − , M n ) for n ≥ . Since the man wins, there exists α > such that lim n →∞ D n = D + α . Moreover, lim n →∞ d ( L n , M n − ) = α . For the geodesic triangle ∆( L n , M n − , M n ) ,consider a comparison triangle ∆( L n , M n − , M n ) in R and denote its interior angle at L n by α n . As d ( M n − , M n ) ≤ D for all n ≥ , it follows that lim n →∞ α n = 0 . Because α n ≤ α n for all n ≥ , we have lim n →∞ α n = 0 . This implies lim n →∞ β n = π as π ≤ β n + α n .Let k > . Then there exists n k ∈ N such that for all n ≥ n k , β n +1 ≥ π − π ⌈ k/D ⌉ . (5)Define γ : [0 , ∞ ) → A , γ ( t ) = (cid:18) − t − nDD (cid:19) L n k + n + t − nDD L n k + n +1 , for nD ≤ t < ( n + 1) D, where n ∈ N . Note that γ ( nD ) = L n k + n for all n ∈ N . We show that γ is a k -local √ -quasi-geodesic ray. Applying thetriangle inequality, d ( γ ( s ) , γ ( t )) ≤ | s − t | ≤ √ | s − t | , for all s, t ≥ . Thus, we only need to prove the following property. Claim 2.
For all s, t ≥ with | s − t | ≤ k , d ( γ ( s ) , γ ( t )) ≥ √ | t − s | . Proof of Claim 2. n ≥ n k , by (5), (cid:18)(cid:24) kD (cid:25) − (cid:19) π + 3 π (cid:24) kD (cid:25) (cid:18) π − π ⌈ k/D ⌉ (cid:19) ≤ ⌈ k/D ⌉ X i =1 β n + i ≤ (cid:24) kD (cid:25) π. For n ≥ n k take B n = ⌈ k/D ⌉ X i =1 β n + i − (cid:18)(cid:24) kD (cid:25) − (cid:19) π. Then π/ ≤ B n ≤ π and so cos B n ≤ −√ / .Let s, t ≥ such that < t − s ≤ k . Fix n ∈ N such that nD ≤ s < ( n + 1) D and nD < t < ( n + 1 + ⌈ k/D ⌉ ) D . Denote m = n k + n and B = B m . Then γ ( s ) ∈ [ L m , L m +1 ] with s = L m +1 , cos B ≤ − √ . (6)and B ≤ β m +1 + . . . + β m + i − ( i − π, for all i ∈ { , . . . , ⌈ k/D ⌉} .We prove Claim 2 by showing that d ( γ ( s ) , γ ( t )) ≥ | cos B | ( t − s ) . Depending on the value of t wedistinguish several situations. For clarity, in each case we will index t .Case I: nD < t ≤ ( n + 1) D . Denote t = t .Then d ( γ ( s ) , γ ( t )) = t − s ≥ | cos B | ( t − s ) .Case II: ( n + 1) D < t ≤ ( n + 2) D . Denote t = t .Let b = d ( γ ( s ) , L m +1 ) = ( n + 1) D − s , c = d ( γ ( t ) , L m +1 ) = t − ( n + 1) D and A = ∠ L m +1 ( γ ( s ) , L m +2 ) = β m +1 ≥ B. Consider a triangle ∆( x, y, z ) in R so that k x − y k = b , k x − z k = c , and the interior angle at x equals A . Since X is a CAT(0) space, d ( γ ( s ) , γ ( t )) ≥ k y − z k . Applying the cosine law in R we get k y − z k = b + c − b c cos A ≥ b + c − b c cos B = b + c + 2 b c | cos B | by (6) ≥ | cos B | ( b + c ) = | cos B | ( t − s ) ≥ | cos B | ( t − s ) . Hence, d ( γ ( s ) , γ ( t )) ≥ | cos B | ( t − s ) .Case III: In general, assume ⌈ k/D ⌉ > and suppose that for i ∈ { , . . . , ⌈ k/D ⌉ − } , A i = ∠ L m + i ( γ ( s ) , L m + i +1 ) ≥ β m +1 + · · · + β m + i − ( i − π and if ( n + i ) D < t i ≤ ( n + i + 1) D , d ( γ ( s ) , γ ( t i )) ≥ | cos B | ( t i − s ) . (7)We show that A i +1 = ∠ L m + i +1 ( γ ( s ) , L m + i +2 ) ≥ β m +1 + · · · + β m + i +1 − iπ and if ( n + i + 1) D < t i +1 ≤ ( n + i + 2) D , d ( γ ( s ) , γ ( t i +1 )) ≥ | cos B | ( t i +1 − s ) . Because β m + i +1 ≤ A i +1 + ∠ L m + i +1 ( γ ( s ) , L m + i ) ≤ A i +1 + π − A i we have A i +1 ≥ β m + i +1 + A i − π ≥ β m +1 + · · · + β m + i +1 − iπ ≥ B. b i +1 = d ( γ ( s ) , L m + i +1 ) ≥ | cos B | (( n + i + 1) D − s ) , where the last inequality follows by applying (7)with t i = ( n + i + 1) D . Take also c i +1 = d ( γ ( t i +1 ) , L m + i +1 ) = t i +1 − ( n + i + 1) D . Consider a triangle ∆( x, y, z ) in R so that k x − y k = b i +1 , k x − z k = c i +1 , and the interior angle at x equals A i +1 . Then d ( γ ( s ) , γ ( t i +1 )) ≥ k y − z k = b i +1 + c i +1 − b i +1 c i +1 cos A i +1 ≥ b i +1 + c i +1 − b i +1 c i +1 cos B = b i +1 + c i +1 + 2 b i +1 c i +1 | cos B | ≥ b i +1 + | cos B | c i +1 + 2 b i +1 c i +1 | cos B | = ( b i +1 + | cos B | c i +1 ) ≥ | cos B | ( t i +1 − s ) . Hence, d ( γ ( s ) , γ ( t i +1 )) ≥ | cos B | ( t i +1 − s ) . This finishes the proof of the claim.We can now state our main result. Theorem 4.3.
Let A be a nonempty, closed and convex subset of a complete CAT(0) space that is additionally δ -hyperbolic. Then the following statements are equivalent:(i) A is geodesically bounded;(ii) A is directionally bounded;(iii) A has the AFPP (for nonexpansive mappings);(iv) A has the FPP (for nonexpansive mappings);(v) the lion always wins the Lion-Man game played in A .Proof. ( i ) ⇐⇒ ( ii ) : Follows from Proposition 3.2. ( ii ) ⇐⇒ ( iii ) : Follows from [34, Theorem 2.4]. ( i ) ⇐⇒ ( iv ) : Follows from [29, Theorem 3.1]. ( v ) = ⇒ ( ii ) : Follows from Theorem 4.1. ( i ) = ⇒ ( v ) : Suppose that A is geodesically bounded and that the man wins. Let k > √ M , where M = M ( δ, √ is given by Remark 3.4. By Proposition 4.2, A contains a k -local √ -quasi-geodesic ray.Using Corollary 3.7 we obtain that A contains a geodesic ray, a contradiction. Final remarks and conclusions
1. In [23] we proved that in the setting of complete, locally compact, uniquely geodesic spaces, if the Lion-Man game is played in a closed and strongly convex domain A , then the lion always wins if and only if A has the fixed point property for continuous mapping (i.e., every continuous self-mapping defined on A hasat least one fixed point). This equivalence is no longer true if the local compactness assumption is droppedbecause, on the one hand, if the Lion-Man game is played in a convex and bounded subset of a Hilbert space,then the lion always wins (see [21]). On the other hand, the unit ball of a Hilbert space has the fixed pointproperty for continuous mappings if and only if the space is finite dimensional.Regarding the relation between the FPP (for nonexpansive mappings) and the solution of the Lion-Mangame, it would be interesting to know if the δ -hyperbolic condition could be removed from Theorem 4.3. Afirst approach to this problem might be to consider the Hilbert framework. Recall that, according to [30],boundedness is a necessary and sufficient condition for a closed and convex subset of a Hilbert space to havethe FPP. Thus we can raise the following problem. Question 1.
If the lion always wins the Lion-Man game played in a closed and convex subset A of a Hilbertspace, must A be bounded?
2. In this paper we considered an ε -capture criterion. Even in compact and convex subsets of the Euclideanplane, there exist games of this type where the lion wins by satisfying the condition D n > D for all n ∈ N and lim n →∞ D n = D (see [23, Example 1]). Thus, for physical capture (i.e., the case when there exists n ∈ N such that D n ≤ D ) we must assume some very rigid geometric conditions. This is the case whenthe domain of the game is a convex and geodesically bounded subset of an R -tree.13o see this, suppose that D n > D for all n ∈ N . Note first that d ( L n , L n +1 ) = D for all n ∈ N . We showthat for every n ∈ N , L n ∈ [ L , L n +1 ] . (8)In this case, L i ∈ [ L , L n ] for all n ∈ N and all i ≤ n . Therefore, d ( L , L n ) = nD for all n ∈ N and S n ≥ [ L , L n ] is a geodesic ray in A , which is a contradiction.To prove that (8) holds we use an inductive argument. For n = 0 this is obvious. We suppose now that(8) holds for n = k and prove that it also holds for n = k + 1 .First note that as L k ∈ [ L , L k +1 ] and L k +1 ∈ [ L k , M k ] , we have L k , L k +1 ∈ [ L , M k ] . Take now y ∈ A such that [ M k , L ] ∩ [ M k , M k +1 ] = [ M k , y ] . If L k +1 ∈ [ M k , y ] , then L k +1 ∈ [ M k , M k +1 ] and so d ( L k +1 , M k +1 ) ≤ d ( M k , M k +1 ) ≤ D , which contradicts the assumption that D k +1 > D . Thus, L k +1 ∈ [ L , y ] .It is easy to see that y ∈ [ L , M k +1 ] . Otherwise, if [ L , y ] ∩ [ y, M k +1 ] = [ y, z ] for zome z ∈ A , z = y , then z ∈ [ M k , L ] ∩ [ M k , M k +1 ] = [ M k , y ] . This is a contradiction because z ∈ [ y, M k +1 ] .Because L k +1 ∈ [ L , y ] , we get L k +1 ∈ [ L , M k +1 ] . Recalling that L k +2 ∈ [ L k +1 , M k +1 ] we obtain (8) for n = k + 1 . Question 2.
What other geometric conditions imply physical capture?
This work was partially supported by DGES (Grant MTM2015-65242-C2-1P).
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