HHahn-Banach for metric functionals and horofunctions
Anders Karlsson ∗ November, 2019
Abstract
It is observed that a natural analog of the Hahn-Banach theorem is valid for metricfunctionals but fails for horofunctions. Several statements of the existence of invariantmetric functionals for individual isometries and 1-Lipschitz maps are proved. Variousother definitions, examples and facts are pointed out related to this topic. In particular itis shown that the metric (horofunction) boundary of every infinite Cayley graphs containsat least two points.
It is well-recognized that the Hahn-Banach theorem concerning extensions of continuous linearfunctionals is a cornerstone of functional analysis. Its origins can be traced to so-called momentproblems, and in addition to H. Hahn and Banach, Helly’s name should be mentioned in thiscontext ([Di81, P07]).In an important recent development, called the
Ribe program, notions from geometric Banachspace theory are formulated purely in terms of the metric associated with the norm, and thenstudied for metric spaces. This has been developed by Bourgain, Ball, Naor and others, for arecent partial survey, see [N18]. This subject is in part motivated by significant applicationsin computer science.It has been remarked in a few places (see [Y11, Ka02] for merely two references, the earliestdiscussion would surely be Busemann’s work in the 1930s, see [Pa05] for a good exposition ofBusemann’s work in this context) that Busemann functions or horofunctions serve as replace-ment for linear functions when the space is not linear. This has been a useful notion in thetheory of manifolds with non-negative curvature as well as for spaces of non-positive curva-ture, and more recently several other contexts beyond any curvature restriction, for example[LN12, LRW18, W18, W19, KaL07]. In parallel, they have been identified or described formore and more metric spaces, see [Ka19] for references.In the present paper, we continue to consider the analogy in this direction between the lineartheory and metric theory. More precisely, we consider the category of metric spaces and semi-contractions (non-expanding maps or -Lipschitz maps), that is, maps f : X → Y which donot increase distances: d ( f ( x ) , f ( x )) ≤ d ( x , x ) for all x , x ∈ X . As discussed below one has metric analogs of the norm, spectral radius,weak topology, the Banach-Alaoglu theorem, and the spectral theorem. The present paper ∗ Supported in part by the Swiss NSF. a r X i v : . [ m a t h . M G ] J a n ocuses on pointing out an analog of the Hahn-Banach theorem. To our mind, the circumstancethat fruitful such analogs are present is surprising and promising.We follow Banach in his paper [Ba25] and in his classic text The Theory of Linear Operators from 1932 in calling maps from a metric space X into R functionals. Since we in additionconsider semi-contractions as our morphisms, we prefix the functionals that we will considerby the word metric . These metric functionals generalize Busemann functions and horofunc-tions, see the next section for precise definitions of the latter two concepts, and are moreoveran analog and replacement for continuous linear functionals in standard functional analysis,perhaps further motivating the use of the word functional .To be precise, with a fixed origin x ∈ X , the following functions and their limits in thetopology of pointwise convergence are called metric functionals : h x ( · ) := d ( · , x ) − d ( x , x ) . The closure of this continuous injection of X into functionals, is called the metric compact-ification of X and is compact by the analog of Banach-Alaoglu, see below. If X is properand geodesics we call the subspace of new points obtained with the closure the metric bound-ary. In discussions with Tobias Hartnick, we observed the following statement (and as will beexplained the corresponding statement for horofunctions is false).
Proposition. (Metric Hahn-Banach statement) Let ( X, d ) be a metric space with base point x and Y a subset containing x . Then for every metric functional h of Y there is a metricfunctional H of X which extends h in the sense that H | Y = h . Because the above is a direct consequence of compactness, it is a bit artificial to speak ofapplications of it in a strict sense. However, it does provide a point of view that we believe isuseful.Note that general Cayley graphs are highly non-trivial metric spaces, so the following state-ment is not an obvious fact:
Theorem.
Let Γ be a finitely generated, infinite group and X a Cayley graph defined by Γ and a finite set of generators. Then the metric boundary of X contains at least two points,and each metric functional is unbounded. Fixed point theorems are of fundamental importance in analysis. The following results arerelated to fixed point statements:
Theorem.
For any monotone distorted isometry g of a metric space X , there exists a metricfunctional that vanishes on the whole orbit { g n x : n ∈ Z } . Thus this result shows that the classical picture of parabolic isometries preserving horospheresin hyperbolic geometry extends, see Figure 1. These two theorems are proved in section 5 wherealso some further results about metric functionals of general Cayley graphs are discussed.With Bas Lemmens we observed the following improvement of the theorem but under anotherhypothesis on the map:
Proposition.
Suppose that f is a semi-contraction of a metric space X with inf x d ( x, f ( x )) =0 . Then there is a metric functional h, such that h ( f ( x )) ≤ h ( x ) for all x ∈ X . In case f is an isometry the inequality is an equality. f is an isometry (with the alternative assumptions)then h ( f n x ) = h ( x ) = 0 for all n .Bader and Finkelshtein defined a reduced boundary by identifying functions in the metricboundary if they differ by a bounded function [BF19]. We note that: Proposition.
Every isometry of a metric space fixes a point in the reduced metric compacti-fication.
Busemann wrote in 1955 that “... two startling facts: much of Riemannian geometry is nottruly Riemannian and much of differential geometry requires no derivatives”. Although itwould be too much to affirm that much of functional analysis requires no linear structure, atleast there are a number of analogs for general metric spaces.
Acknowledgements:
It is a pleasure to thank Tobias Hartnick for the invitation to Giessenand our discussion there that sparked the origin of this note. I also thank Nate Fisher, Pierrede la Harpe, and Bas Lemmens for useful discussions.
Let ( X, d ) be a metric space. One defines (a variant of the map considered by Kuratowskiand Kunugui in the 1930s, see for example [L93, p. 45]) Φ : X → R X via x (cid:55)→ h x ( · ) := d ( · , x ) − d ( x , x ) . As the notation indicates the topology we take here in the target space is pointwise conver-gence. The map is continuous and injective.
Proposition 1. (Metric Banach-Alaoglu) The space Φ( X ) is a compact Hausdorff space.Proof. By the triangle inequality we note that − d ( x , y ) ≤ h x ( y ) ≤ d ( y, x ) , which implies that the closure Φ( X ) is compact by the Tychonoff theorem. It is Hausdorff,since it is a subspace of a product space of Hausdorff spaces, indeed metric spaces.3n general this is not a compactification of X in the strict and standard sense that the spaceis embedded, but it is convenient to still call it a compactification. In other words, we providea weak topology that has compactness properties.By the triangle inequality it follows that all the elements in X := Φ( X ) are semi-contractivefunctionals X → R , and we call them metric functionals . Example 2.
Let γ be a geodesic line (or just a ray γ : R + → X ), which is a standard notionin metric geometry at least since Menger. Then the following limit exists: h γ ( y ) = lim t →∞ d ( y, γ ( t )) − d ( γ (0) , γ ( t )) . The reason for the existence of the limit for each y is that the sequence in question is boundedfrom below and monotonically decreasing (thanks to the triangle inequality), see [BGS85,BrH99]. This element in Φ( X ) is called the Busemann function associated with γ . Example 3.
The open unit disk of the complex plane admits the Poincaré metric, which inits infinitesimal form is given by ds = 2 | dz | − | z | . This gives a model for the hyperbolic plane and moreover every holomorphic self-map of thedisk is a semi-contraction in this metric (the Schwarz-Pick lemma). The Busemann functionassociated to the (geodesic) ray from 0 to the boundary point ζ , in other words ζ ∈ C with | ζ | = 1 , is h ζ ( z ) = log | ζ − z | − | z | . These functions appear (in disguise) already in 19th century mathematics, such as in thePoisson integral representation formula and in Eisenstein series.The more common choice of topology, introduced by Gromov in [Gr81], is uniform conver-gence on bounded sets. We denote the corresponding closure X h and call it the horofunctionbordification , the new points are called horofunctions following common terminology. Thisclosure amounts to the same compactification if X is proper (i.e. closed bounded sets arecompact), see [BrH99], but in general it is quite different, in particular there is no notionof weak compactness. As in the example described in Remark 8 below, a space may haveno horofunctions. Moreover, the following example shows that Busemann functions are notalways horofunctions, since the limit above might not converge in this topology: Example 4.
Take one ray [0 , ∞ ] that will be geodesic γ , then add an infinite number of pointsat distance to the point x = 0 and distance to each other. Then at each point n on theray, connect it to one of the points around 0 (that has not already been connected) with ageodesic segment of length n − / . This way h γ ( y ) = lim t →∞ d ( γ ( t ) , y ) − d ( γ ( t ) , γ (0)) stillof course converge for each y but not uniformly. Hence the Busemann function h γ is a metricfunctional but not a horofunction. See Figure 2.The topology that we chose here has been useful in a few instances already: [GV12, GK15,G18, MT18] and in recent work by Bader and Furman.In a Banach space the metric functional associated to points are not linear since they areclosely related to the norm, but sometimes their limits are linear, see for example [Ka19]. Inany case they are all convex functions: 4igure 2: The ray γ does not define a horofunction. Proposition 5.
Let E be a normed vector space. Every function h ∈ E is convex, that is, forany x, y ∈ X one has h ( x + y ≤ h ( x ) + 12 h ( y ) . Proof.
Note that for z ∈ E one has h z (( x + y ) /
2) = (cid:107) ( x + y ) / − z (cid:107) − (cid:107) z (cid:107) = 12 (cid:107) x − z + y − z (cid:107) − (cid:107) z (cid:107)≤ (cid:107) x − z (cid:107) + 12 (cid:107) y − z (cid:107) − (cid:107) z (cid:107) = 12 h z ( x ) + 12 h z ( y ) . This inequality passes to any limit point of such h z .In [GK15, Lemma 3.1] it is shown that for any metric functional h of a real Banach spacethere is a linear functional f of norm at most 1 such that f ≤ h . The proof uses the standardBanach-Alaoglu and Hahn-Banach theorems. Example 6.
Gutiérrez has provided a good description of metric functionals for L p spaces p ≥ . To give an idea we recall the formulas for (cid:96) p ( J ) for < p < ∞ . There are two types: h z,c ( x ) = (cid:16) (cid:107) x − z (cid:107) pp + c p − (cid:107) z (cid:107) pp (cid:17) /p − c, where z ∈ (cid:96) p ( J ) and c ≥ (cid:107) z (cid:107) p , as well as h µ ( x ) = − (cid:88) j ∈ J µ ( j ) x ( j ) , where µ ∈ (cid:96) q ( J ) , with q the conjugate exponent, and (cid:107) µ (cid:107) q ≤ . See [Gu17, Gu18, Gu19] formore details and precise statements. An interesting detail that Gutiérrez showed is that thefunction identically equal to zero is not a metric functional for (cid:96) (in contrast to the space (cid:96) ).He also observed how a famous fixed point free example of Alspach fixes a metric functional.For another discussion about Busemann functions of certain normed spaces, see [W07, W18].5igure 3: A counter-example. Proposition 7. (Metric Hahn-Banach statement.) Let ( X, d ) be a metric space with basepoint x and Y a subset containing x . Then for every h ∈ Y there is a metric functional H ∈ X which extends h in the sense that H | Y = h .Proof. Given h ∈ Y . Since the metric compactifications are Hausdorff (even metrizable if X is separable) we take a net h y α that converges to the unique limit h. These points y α are alsopoints in X and by compactness of X also has a limit point H there. By uniqueness of thelimits it must coincide with h . Remark . On the other hand, for horofunctions, i.e. for X h , the Hahn-Banach theorem does not hold. Example 4 shows this, with Y taken to be the (image of the) geodesic ray γ . Then Y clearly has a metric functional b that is a Busemann function, however in X any sequence ofpoints going to infinity (i.e. along Y ) cannot converge in X h , but to extend b we would needsuch. Another illustration of this phenomenon (a counter-example to the proof but not thestatement) is the following: Consider longer and longer finite closed intervals [0 , n ] all gluedto a point x . See Figure 3. This becomes a countable (metric) tree which is unbounded butcontains no infinite geodesic ray. Denote by x n the other end points of each interval. Thesequence h x n do not converge uniformly on balls. On the other hand it does so in Y beingjust the set { x n } n ≥ .Here is a remark in another direction: Remark . From a more general point of view, one could expect the possibility of extensionsof metric functionals. The real line ( R , |·| ) as a metric space is an injective object in thecategory we consider, namely metric spaces and semi-contractions (i.e. -Lipschitz maps),in the following restricted sense: Given any metric space A and a monomorphism that isisometric φ : A → B where B is another metric space, and semi-contraction f : A → R thereis an extension in the obvious sense of f to a morphism B → R , for example ¯ f ( b ) := sup a ∈ A ( f ( a ) − d ( φ ( a ) , b )) or ¯ f ( b ) := inf a ∈ A ( f ( a ) + d ( φ ( a ) , b )) .
6o see this, in the former definition, note first that for b (cid:48) = φ ( a (cid:48) ) one has f ( a (cid:48) ) − d ( φ ( a (cid:48) ) , φ ( a (cid:48) )) ≥ f ( a ) − d ( φ ( a ) , φ ( a (cid:48) )) , which shows that our map is an extension, and then in general that f ( b ) − f ( b (cid:48) ) ≤ sup a ( f ( a ) − d ( φ ( a ) , b )) − sup a (cid:48) (cid:0) f ( a (cid:48) ) − d ( φ ( a (cid:48) ) , b (cid:48) ) (cid:1) ≤ sup a (cid:0) f ( a ) − d ( φ ( a ) , b ) − f ( a ) + d ( φ ( a ) , b (cid:48) ) (cid:1) ≤ d ( b, b (cid:48) ) , which implies it is a morphism. The origin of this observation is [McS34].There are topological vector spaces with trivial dual. Maybe in spirit this is a bit similar tothe following example: Example 10.
Let D : R ≥ → R ≥ be an increasing function with D (0) = 0 , D ( t ) → ∞ and D ( t ) /t → monotonically. The latter condition implies that D ( t + s ) ≤ D ( t ) + D ( s ) , so onesees that ( R , D ( |·| )) is a metric space. As is observed in [KaMo08], D ( t ) /t → implies that ( R , D ( |·| )) = { h x : x ∈ R } ∪ { h ≡ } . That is, there is a metric functional which vanishes on the whole space, and the compactifica-tion is the one-point compactification. Note that this metric space has no geodesics explainingin particular why there are no Busemann functions.In contrast to the linear theory, note that not every metric functional of X is a metric functionalof a subset Y as the following example illustrates: Example 11.
Let X be the Euclidean space R d and Y = R be a one dimensional linearsubspace. The Busemann function associated to a ray from the origin perpendicular to Y vanishes identically on Y . On the other hand the zero function is not a metric functional on Y . Let f be a semi-contraction of a metric space to itself. As remarked in [Ka19], the minimaldisplacement of f , d ( f ) = inf x d ( x, f ( x )) , is the analog of the norm of a linear operator andthe translation number is the analog of the spectral radius: τ ( f ) = lim n →∞ n d ( x, f n ( x )) , which exists in view of subadditivity. Similar to relationship between the norm and spectralradius, one always has that τ ( f ) ≤ d ( f ) , since the translation number is independent of x .Note that in general the inequality may be strict, for example a rigid rotation of the circle,or more interestingly, for groups with a word metric all non-identity elements g have d ( g ) ≥ ,but can easily have τ ( g ) = 0 .In passing we record the following simple fact. Proposition 12.
The following tracial property holds: τ ( gf ) = τ ( f g ) for any two semi-contractions f and g . roof. From the triangle inequality, d ( x, ( f g ) n ( x )) ≤ d ( x, f ( x )) + d ( f ( x ) , ( f g ) n ( x )) ≤ d ( x, f ( x )) + d ( x, ( gf ) n − g ( x )) ≤ d ( x, f ( x )) + d ( x, ( gf ) n ( x )) + d (( gf ) n ( x ) , ( gf ) n − g ( x )) ≤ d ( x, f ( x )) + d ( x, ( gf ) n ( x )) + d ( f ( x ) , x ) . Dividing by n and sending n to infinity shows one inequality. By symmetry also the oppositeinequality holds.We say that an isometry g is distorted if τ ( g ) = 0 . Example 10 provides an observation relatedto distortion. This example could be considered for Z (instead of R ) and both are groups andthe metric is invariant (but not a word-metric). Here all non-zero elements are distorted.It was shown in [Ka01], see also [Ka19], that for any semi-contraction f there is a metricfunctional h such that h ( f n ( x )) ≤ − τ ( f ) n for all n > and lim n →∞ − n h ( f n ( x )) = τ ( f ) . I will refer to this result as the metric spectral principle . As A. Valette pointed out to me,this could also be viewed as a statement in the spirit of the classical Hahn-Banach theorem:existence of a functional that realizes the norm of particular element.In discussions with Bas Lemmens we observed the following, which strengthen the previousstatement in a special case (another more general strengthening appears in [GV12], but underweak non-positive curvature assumptions on X ): Proposition 13.
Suppose that f is a semi-contraction of X with d ( f ) = 0 . Then there is ametric functional h, such that h ( f ( x )) ≤ h ( x ) for all x ∈ X .Proof. For any (cid:15) > we define the sets N (cid:15) = { x ∈ X : d ( x, f ( x )) ≤ (cid:15) } and note that they are closed sets, and form a nested family. They are all non-empty bythe assumption on f . By compactness their intersection must contain a metric functional h .There is thus a sequence of points x i which converges to h in the usual weak sense (fix asequence of (cid:15) going to and take a point x i in each set). These x i are moved less and less by f which means that h ( f ( x )) = lim i →∞ d ( f ( x ) , x i ) − d ( x , x i ) = lim i →∞ d ( f ( x ) , f ( x i )) − d ( x , x i ) ≤ lim i →∞ d ( x, x i ) − d ( x , x i ) = h ( x ) . orollary 14. Suppose the displacement of an isometry g is zero, then there is an invariantmetric functional h, in the strong sense that h ( gx ) = h ( x ) for all x .Proof. This is immediate from the previous proposition and its proof.Note that this is not in contradiction with a complicated isometry like Edelstein’s example,see [Ka01], since h ≡ is a metric functional for Hilbert spaces.In the following section we will prove similar statements as above in the case when τ ( g ) = 0 but d ( f ) possibly strictly positive. Let me first record some facts that we, and probably others, have realized years ago, see[KaL07, Ka08]. They are however not generally known by people in geometric group theory.Let X be a Cayley graph of a finitely generated group Γ , which becomes a metric spacewith the corresponding word metric associated to a finite generating set, see the recent book[CH16] for a wealth of metric geometry in this setting. For Cayley graphs we always take theneutral element as base point, x = e . The group acts by isometry on X , and this actionsextends continuously to an action by homeomorphism of X and also on the metric boundary ∂X := X \ X. This boundary is a compact metrizable space. Let λ be a Γ -invariant probabilitymeasure on ∂X . Then the following map T ( g ) := (cid:90) ∂X h ( g ) dλ ( h ) is a 1-Lipschitz group homomorphism Γ → R .No metric functional on X is identically zero, see below. Therefore if Γ fixes a h ∈ ∂X thenthere is a non-trivial homomorphism T : Γ → Z . A further idea shown in [Ka08] is thatgiven a finitely generated group with countable boundary, there is a finite index subgroupthat surjects on Z . One might wonder ([Ka08]) or even conjecture ([TY16]) that every groupof polynomial growth has countable boundary. This would immediately imply the celebratedtheorem of Gromov that such groups are virtually nilpotent. Some positive evidence forthis approach is provided in [W11, TY16]. One could moreover have some hope that growthconsiderations in relation to the metric boundary could yield more than what is already knownin this direction (recall that Grigorchuk’s Gap Conjecture remains open).It is well-known that infinite finitely generated groups (and their Cayley graphs) can be verycomplicated, for example it can contain only elements of finite order, or even an exponent N such that g N = 1 for all group elements g . Therefore I think that the following is a non-obviousfact. Theorem 15.
Let Γ be an infinite, finitely generated group and X a Cayley graph associatedto a finite set of generators. Then the metric boundary must contain at least two points andall metric functionals are unbounded, in the sense that for any metric functional h there is noupper bound of | h ( x ) | as a function of x .Proof. First we establish the second assertion. Note that given the assumptions, the graph X is a proper metric space that is geodesic, in particular connected, and with infinite diameter.In view of the latter property, any h g with g ∈ Γ is clearly unbounded. It remains to considera limit function h ( x ) = lim n →∞ d ( x, g n ) − d ( e, g n ) . Since the distance function is integer9alued and the graph locally finite, it holds that for any r > there is a number N suchthat h ( x ) = d ( x, g n ) − d ( e, g n ) for all n ≥ N and x such that d ( x, e ) ≤ r. Take a geodesicfrom e to g N , it must intersect the sphere around e of radius r in a point y . This means that h ( y ) = d ( y, g N ) − d ( g N , e ) = − r . The statement follows since r was arbitrary.Now assume that the metric boundary contains exactly one point, ∂X = { h } . By the remarksabove, h defines a homomorphism Γ → Z by γ (cid:55)→ h ( γ ) . By what has just been shown, h is unbounded, in particular not identically . This has asa consequence that the homomorphisms is non-trivial. This means that there must exist anundistorted, infinite order element g (since any homomorphism into Z must annihilate finiteorder and distorted elements), with h ( g ) (cid:54) = 0 . Without loss of generality we may assume h ( g ) > (otherwise replacing g by its inverse). But now by the metric spectral principlerecalled above, there must exist a metric functional h such that h ( g ) ≤ . This shows that h is different from h . Since τ ( g ) > the functional h must take arbitrarily large negativevalues, showing that h ∈ ∂X .In classical hyperbolic geometry, parabolic isometries are those which are distorted and pre-serves a horosphere. We say that an isometry g of a metric space X with base point x is monotone if d ( x , g n x ) → ∞ monotonically for all sufficiently large n as n → ∞ . (So one wayof treating a general isometry f might be to pass to a power of it, g := f N .) Theorem 16.
Let g be a monotone isometry of a metric space X . If τ ( g ) = 0 , then thereexists a metric functional that vanishes on the whole orbit { g n x : n ∈ Z } .Proof. Let Y be the orbit { g n x : n ∈ Z } with the metric induced by X. It is a proper metricspace since g is monotone. Consider the subset A of ∂Y of metric functionals h for which h ( g n x ) ≤ h ( g m x ) for all n ≥ m. The subset A is closed since the inequalities pass to limits.It is also invariant under the group H generated by g since for any n ≥ m , ( g.h )( g n x ) = h ( g n − x ) − h ( g − x ) = h ( g n − x ) − h ( g m − x ) + ( g.h )( g m x ) ≤ ( g.h )( g m x ) . Next we verify that A is non-empty. Take a converging subsequence of h g n x as n → ∞ , innotation h g ni x → h . Then notice that by the monotonicity of g , for fixed m ≥ k , we have h ( g m x ) = lim i d ( g m x , g n i x ) − d ( x , g n i x )= lim i d ( x, g n i − m x ) − d ( x , g n i x ) ≤ lim i d ( x , g n i − k x ) − d ( x , g n i x ) = h ( g k x ) . Since H is a cyclic group acting on the compact non-empty set A by homeomorphisms, thereis an invariant probability measure µ on A . Therefore, as remarked above (with details foundin [Ka08, Proposition 2]), T ( g ) = (cid:90) ∂Y h ( gx ) dµ ( h ) defines a -Lipschitz group homomorphism T : H → R . Since any element of R is undistortedthe image of g must be , and so for every n >
00 = (cid:90) ∂Y h ( g n x ) dµ ( h ) .
10n the other hand h ( g n x ) ≤ h ( x ) = 0 . This implies that for every n the set of h forwhich h ( g n x ) = 0 has full measure. The intersection of countable full measure sets has fullmeasure, therefore there exists at least one h which vanishes on the whole orbit, thus provingthe theorem. Remark . Note that the function h ≡ is a metric functional on any infinite dimensionalHilbert space, while as an additive group no element is distorted.Now we observe one thing in relation to the interesting notion of reduced boundary from[BF19] (note a conjecture in this paper that states that for finitely generated nilpotent groupsall reduced boundary points should be fixed by the whole group). We extend their definitionby also considering points in X and in the weak topology. It is easy to verify that the actionof isometries extends to the reduced metric boundary, with the following calculation: Say that H is equivalent to h differing by at most a constant C . Then | gH ( x ) − gh ( x ) | = (cid:12)(cid:12) H ( g − x ) − H ( g − x ) − h ( g − x ) + h ( g − x ) (cid:12)(cid:12) ≤ C. Proposition 18.
Let g be an isometry of a metric space. Then it fixes a point in the reducedmetric compactification.Proof. Take any limit point h of the orbit g n x which exists by compactness. Recall that ( g.h )( x ) = h ( g − x ) − h ( g − x ) . The last term is bounded so we can forget this when passingto the reduced compactification. We calculate: h ( g − x ) = lim i →∞ d ( g − x, g n i x ) − d ( x , g n i x ) ≤ lim inf i d ( g − x, g n i − x ) + d ( g n i − x , g n i x ) − d ( x , g n i x )= d ( g − x , x ) + lim i d ( x, g n i x ) − d ( x , g n i x ) = d ( g − x , x ) + h ( x ) , for any x ∈ X . The reverse inequality is obtained by applying the inequality to gx insteadof x. Since the action by the isometry g is a well-defined map of the reduced boundary, thisproves the proposition. This short sections provide some remarks and suggestions of preliminary nature. Recall thatin Banach spaces weakly convergent sequences are bounded. Also if a weakly convergentsequence belong to a closed convex set, then the weak limit is also contained in this set(Mazur’s theorem, see for example [La02, p. 103]). Notice that in the example in Remark 8, h x n converges weakly but is not bounded. On the other hand the limit belongs to the convexhull of the sequence. One could wonder if there is a generalization of this. In a yet differentdirection related to this, we refer the reader to [Mo16].One could imagine defining the weak topology by declaring a sequence x n weakly convergingto x if h ( x n ) → h ( x ) for every metric functional on X. But this would give back the the usualstrong, or metric, topology, since we could look at h = h x . Then it would make more sense toonly consider h which are at infinity. (I am indebted to V. Guirardel and T. Hartnick for theseremarks.) Now it could be interesting to see in what way these h coordinates the space x. for11xample if X is a geodesic space moreover with the property that every geodesic segment canbe extended to a ray, then one would have d ( x , x ) = sup h | h ( x ) | , where the supremum is taken over all Busemann functions.One could also try to define a norm on the Busemann functions (or metric functionals comingfrom unbounded sequences): (cid:107) h (cid:107) = sup x (cid:54) = x | h ( x ) | d ( x, x ) . Note that this is not always , for example for infinite dimensional Hilbert spaces where alllinear functionals of norm at most 1 are metric functionals, see [Ka19]. With this one wouldhave a metric on the “dual space” of a metric space. At times, I have also suggested the notionfor a metric space to be reflexive: if every horofunction is a Busemann function.Here is a remark from [Ka19]: In the works of Cheeger and collaborators on differentiabilityof functions on metric spaces, the notion of generalized linear function appears. In [Ch99] itis connected to Busemann functions, on the other hand that author remarks in [Ch12] thatnon-constant such functions do not exists for most spaces. Perhaps it remains to investigatehow metric functionals relate to this subject.It is shown above (and easily observed in case of abelian groups for example) that the metricboundary always contains at least two points. One may therefore conclude that is is notthe Poisson boundary of random walks. But it conceivable at least for some large classes ofgroups, that if we pass to the reduced boundary, then random walks could converge to pointsin this space (recurrent, or drift zero, random walk could be said to converge to the interiorwhich reduced is one point). In this way hitting probability measures on the reduced metricboundary could describe the behavior of random walks in a more refined way. For example,random walks on the integers is governed by the classical law of large numbers under a firstmoment condition. The expectation value can be negative, zero or positive, depending on the ifthe defining random walk measure is asymmetric. The reduced metric compactification (withrespect to any generating set) consists of three points, that naturally can be denoted −∞ , + ∞ and all finite points, that is, Z itself. And these three points describe the asymptotic behaviorof random walks with negative, positive, or no drift, respectively. If the reduced boundaryis not enough one could consider the star geometry, in the sense of [Ka05], associated to the(reduced) metric compactification. This could be invariant of the chosen generating set for theCayley graphs. It would also be of interest to study how groups act on this, their associatedincidence geometry. This is related to the conjecture in [BF19] already mentioned above. References [BF19] Bader, Uri, Finkelshtein, Vladimir, On the horofunction boundary of discreteHeisenberg group, preprint 2019, arXiv:1904.11234[BGS85] Ballmann, Werner; Gromov, Mikhael; Schroeder, Viktor, Manifolds of nonpositivecurvature. Progress in Mathematics, 61. Birkhäuser Boston, Inc., Boston, MA,1985. vi+263 pp.[Ba25] Banach, S. Sur le prolongement de certaines fonctionnelles Bulletin des ScencesMatheumatiques (2) 49, 1925, pages 301-307.12BrH99] Bridson, Martin R.; Haefliger, André, Metric spaces of non-positive curva-ture. Grundlehren der Mathematischen Wissenschaften [Fundamental Principlesof Mathematical Sciences], 319. Springer-Verlag, Berlin, 1999. xxii+643 pp.[Ch12] Cheeger, Jeff, Quantitative differentiation: a general formulation. Comm. PureAppl. Math. 65 (2012), no. 12, 1641–1670.[Ch99] Cheeger, J. Differentiability of Lipschitz functions on metric measure spaces. Geom.Funct. Anal. 9 (1999), no. 3, 428–517.[Cl18] Claassens, Floris, The horofunction boundary of infinite dimensional hyperbolicspaces, arxiv preprint 2018[CH16] Cornulier, Yves; de la Harpe, Pierre, Metric geometry of locally compact groups.EMS Tracts in Mathematics, 25. European Mathematical Society (EMS), Zürich,2016. viii+235 pp.[Di81] Dieudonné, Jean History of functional analysis. North-Holland Mathematics Stud-ies, 49. Notas de Matemática [Mathematical Notes], 77. North-Holland PublishingCo., Amsterdam-New York, 1981. vi+312 pp.[GV12] Gaubert, S. Vigeral, G. 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J. Eur. Math. Soc. [Gr81] Gromov, M. Hyperbolic manifolds, groups and actions. Riemann surfaces and re-lated topics: Proceedings of the 1978 Stony Brook Conference (State Univ. NewYork, Stony Brook, N.Y., 1978), pp. 183–213, Ann. of Math. Stud., 97, PrincetonUniv. Press, Princeton, N.J., 1981.[Gu17] Gutiérrez, Armando W. The horofunction boundary of finite-dimensional (cid:96) p spaces,Colloq. Math. 155 (2019), no. 1, 51–65[Gu18] Gutiérrez, Armando W. On the metric compactification of infinite-dimensional Ba-nach spaces,
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62 (2019), no. 3, 491–507.[Gu19] Gutiérrez, Armando W. The metric compactification of L p represented by randommeasures, to appear in Annals of Functional Analysis. [Ka01] Karlsson, Anders, Non-expanding maps and Busemann functions. Ergodic TheoryDynam. Systems 21 (2001), no. 5, 1447–1457.[Ka02] Karlsson, Anders Nonexpanding maps, Busemann functions, and multiplicativeergodic theory. Rigidity in dynamics and geometry (Cambridge, 2000), 283–294,Springer, Berlin, 2002.[Ka05] Karlsson, Anders, On the dynamics of isometries. Geom. Topol. 9 (2005),2359–2394. 13KaL06] Karlsson, Anders; Ledrappier, François, On laws of large numbers for randomwalks. Ann. Probab. 34 (2006), no. 5, 1693–1706.[KaL07] Karlsson, Anders; Ledrappier, François Linear drift and Poisson boundary for ran-dom walks. Pure Appl. Math. Q. 3 (2007), no. 4, Special Issue: In honor of GrigoryMargulis. Part 1, 1027–1036.[Ka08] Karlsson, Anders, Ergodic theorems for noncommuting random products, lecturenotes available online since 2008[KaL11] Karlsson, Anders; Ledrappier, François, Noncommutative ergodic theorems.
Ge-ometry, rigidity, and group actions , 396–418, Chicago Lectures in Math., Univ.Chicago Press, Chicago, IL, 2011,[KaMo08] Karlsson, Anders, Monod, Nicolas, Strong law of large numbers with concave mo-ments, unpublished note 2008[Ka19] Karlsson, Anders, Elements of a metric spectral theory, To appear in a volume inhonor of Margulis, 2019[L93] Lang, Serge Real and functional analysis. Third edition. Graduate Texts in Math-ematics, 142. Springer-Verlag, New York, 1993. xiv+580 pp.[La02] Lax, Peter D. Functional analysis. Pure and Applied Mathematics. Wiley-Interscience, New York, 2002. xx+580 pp.[LN12] Lemmens, Bas; Nussbaum, Roger Nonlinear Perron-Frobenius theory. CambridgeTracts in Mathematics, 189. Cambridge University Press, Cambridge, 2012. xii+323pp.[LRW18] Lemmens, Bas; Roelands, Mark; Wortel, Marten, Isometries of infinite dimensionalHilbert geometries. J. Topol. Anal. 10 (2018), no. 4, 941–959.[MT18] Maher, Joseph, Tiozzo, Giulio, Random walks on weakly hyperbolic groups, toappear in
Journal für die reine und angewandte Mathematik 2018. [McS34] McShane, E. J. Extension of range of functions. Bull. Amer. Math. Soc. 40 (1934),no. 12, 837–842.[Mo16] Monod, Nicolas, Extreme points in non-positive curvature. Studia Math. 234(2016), no. 3, 265–270.[N18] Naor, Assaf, Metric dimension reduction: A snapshot of the Ribe program, toappear in Proceedings of the ICM 2018 https://arxiv.org/abs/1809.02376[Pa05] Papadopoulos, Athanase Metric spaces, convexity and nonpositive curvature.IRMA Lectures in Mathematics and Theoretical Physics, 6. European Mathemat-ical Society (EMS), Zürich, 2005. xii+287 pp.[P07] Pietsch, Albrecht,