ABSOLUTELY MINIMAL LIPSCHITZ EXTENSION OF TREE-VALUED MAPPINGS

ASSAF NAOR AND SCOTT SHEFFIELD

Abstract. We prove that every Lipschitz function from a subset of a locally compact length space to a metric tree has a unique absolutely minimal Lipschitz extension (AMLE). We relate these extensions to a stochastic game called Politics — a generalization of a game called Tug of War that has been used in [42] to study real-valued AMLEs.

1. Introduction

For a pair of metric spaces (X, d_{X}) and (Z, d_{Z}), a mapping h : X → Z, and a subset S ⊆ X, the Lipschitz constant of h on S is denoted

Lip_{S}(h) =^{def} sup_{\begin{matrix}x,y∈S \\ x≠y \end{matrix}} \frac{d_{Z}(h(x), h(y))}{d_{X}(x, y)}.

Given a closed subset Y ⊆ X and a Lipschitz mapping f : Y → Z, a Lipschitz mapping f̃ : X → Z is called an absolutely minimal Lipschitz extension (AMLE) of f if its restriction to Y coincides with f, and for every open subset U ⊆ X ∖ Y and every Lipschitz mapping h : X → Z that coincides with f̃ on X ∖ U we have

Lip_{U}(h) ⩾ Lip_{U} \binom{}{f} .  (1)

In other words, f̃ extends f, and it is not possible to modify f̃ on an open set in a way that decreases the Lipschitz constant on that set.

Our main result is:

Theorem 1. Let X be a locally compact length space and let T be a metric tree. For every closed subset Y ⊆ X, every Lipschitz mapping f : Y → T has a unique AMLE f̃: X → T.

Recall that a metric space (X, d_{X}) is a length space if for all x, y ∈ X, the distance d_{X}(x, y) is the infimum of the lengths of curves in X that connect x to y. By a metric tree we mean the one-dimensional simplicial complex associated to a finite graph-theoretical tree with arbitrary edge lengths (i.e., a finite graph-theoretical tree whose edges are present as actual intervals of arbitrary length, equipped with the graphical shortest path metric). We did not investigate here the greatest possible generality in which Theorem 1 holds true; in particular, we conjecture that the assumption that X is locally compact can be dropped, and that T need not correspond to a finite graph-theoretical tree, but rather can belong to the more general class of bounded ℝ-trees (see [14, 15]). The requirement that X be locally compact is not used in our proof of the uniqueness assertion of Theorem 1.

In the special case when T is an interval [a, b] ⊆ ℝ and X = ℝ^{n}, Theorem 1 was proved in [17]; see also [6, 4, 2] for different proofs of the uniqueness part of Theorem 1 in this special case. The existence part of Theorem 1 was generalized to arbitrary length spaces X and T = [a, b] in [37]; see also [21] and [29] for different proofs of this existence result with the additional assumptions that the length space X is separable or compact, respectively. The uniqueness part of Theorem 1 was proved in [42] for X a general length space and T = [a, b]. Additionally, [42] contains a new (game theoretic) proof of the existence part of Theorem 1 when T = [a, b] and X is a general length space.

The purpose of the present article is to initiate the study of absolutely minimal Lipschitz extensions of mappings that are not necessarily real-valued, the tree-valued case being the first non-trivial setting of this type where such theorems can be proved. Our proofs overcome various difficulties that arise since we can no longer use the order structure of the real line, which was crucially used in [17, 37, 21, 29, 42]. We also introduce a stochastic game called Politics, related to tree-valued AMLE, that generalizes the stochastic game called Tug of War that was introduced and related to real-valued AMLE in [42].

In the remainder of this introduction we explain the relevant background from the classical theory of Lipschitz extension and ∞-harmonic functions, and also describe the main steps of our proof.

1.1. Background on the Lipschitz extension problem. The classical Lipschitz extension problem asks for conditions on a pair of metric spaces (X, d_{X}) and (Z, d_{Z}) which ensure that there exists K ∈ [1, ∞) such that for all Y ⊆ X and all Lipschitz mappings f : Y → Z, there exists f̃: X → Z with f̃|_{Y} = f and

Lip_{X} \binom{}{f} ⩽ K · Lip_{Y} (f).  (2)

Stated differently, in the Lipschitz extension problem we are interested in geometric conditions ensuring the existence of f̃: X → Z such that the diagram in (3) commutes, where ι : Y → X is the formal inclusion, and the Lipschitz constant of f̃ is guaranteed to be at most a fixed multiple (depending only on the geometry of the spaces X, Z) of the Lipschitz constant of f.

\begin{matrix}X \\ Y & ^{ι } & \overline{f ^{}} & Z & ^{f̃  ^{}} \end{matrix}(3)

Note that if (Z, d_{Z}) is complete then we can trivially extend f to the closure of Y.

When K = 1 in (2), i.e., when one can always extend functions while preserving their Lipschitz constant, the pair (X, Z) is said to have the isometric extension property. When K ∈ (1, ∞) the corresponding extension property is called the isomorphic extension property. The present article is devoted to the isometric extension problem, though we will briefly discuss questions related to its isomorphic counterpart in Section 1.4. We refer to the books [47, 7] and the references therein, as well as the introductions of [30, 40] (and the references therein), for more background on the Lipschitz extension problem.

It is rare for a pair of metric spaces (X, Z) to have the isometric extension property. A famous instance when this does happen is Kirszbraun’s extension theorem [24], which asserts that if X and Z are Hilbert spaces then (X, Z) have the isometric extension property.

Another famous example is the non-linear Hahn-Banach theorem [34], i.e., when Z = ℝ and X is arbitrary; this (easy) fact follows from the same proof as the proof of the classical Hahn-Banach theorem (i.e., by extending to one additional point at a time; alternatively, one can construct the maximal and minimal isometric extensions explicitly).

More generally, one may consider metric spaces Z such that for every metric space X the pair (X, Z) has the isometric extension property (i.e., Z is an injective metric space in the isometric category). This is equivalent to the fact that there is a 1-Lipschitz retraction from any metric space containing Z onto Z (see [7, Prop. 1.2]); such spaces are called in the literature absolute 1-Lipschitz retracts. It is a well known fact (see [7, Prop. 1.4]) that (Z, d_{Z}) is an absolute 1-Lipschitz retract if and only if (a) Z is metrically convex, i.e., for every x, y ∈ Z and λ ∈ [0, 1] there is z ∈ Z such that d_{Z}(x, z) = λd_{Z}(x, y) and d_{Z}(y, z) = (1 − λ)d_{Z}(x, y), and (b) Z has the binary intersection property, i.e., if every collection of pairwise intersecting closed balls in Z has a common point. Examples of absolute 1-Lipschitz retracts are ℓ_{∞} and metric trees (see [23, 19]). Additional examples are contained in [16] (see also [7, Ch. 1]).

If (X, d_{X}) is path-connected and the pair (X, Z) has the isometric extension property, then the AMLE condition (1) is equivalent to the requirement:

∀ open U ⊆ X ∖ Y, Lip_{U} \binom{}{f} = Lip_{∂U} \binom{}{f} .  (4)

When Z = ℝ and X = ℝ^{n}, the AMLE formulation (4) was first introduced by Aronsson [3], in connection with the theory of ∞-harmonic functions. Specifically, it was shown in [3] that if f̃: ℝ^{n} → ℝ is smooth then the validity of (4) is equivalent to the requirement that

∑_{i=1}^{n} ∑_{j=1}^{n} \frac{∂f̃}{∂x_{i}} · \frac{∂f̃}{∂x_{j}} · \frac{∂^{2}f̃}{∂x_{i}∂x_{j}} = 0 on ℝ^{n} ∖ Y.  (5)

If one interprets (5) in terms of viscosity solutions, then it was proved in [17] that the equivalence of (4) and (5) (when Z = ℝ and X = ℝ^{n}) holds for general Lipschitz f̃. We refer to the survey article [4] and the references therein for more information on the many works that investigate this remarkable connection between the classical Lipschitz extension problem and PDEs.

Existence of isometric and isomorphic Lipschitz extensions has a wide variety of applications in pure and applied mathematics. Despite this rich theory, the issue raised by Aronsson’s seminal paper [3] is that even when isometric Lipschitz extension is possible, many such extensions usually exist, and it is therefore natural to ask for extension theorems ensuring that the extended function has additional desirable properties. In particular, the notion of AMLE is an isometric Lipschitz extension which is locally the “best possible” extension. In this context, one can ask for (appropriately defined) “AMLE versions” of known Lipschitz extension theories. As a first step, in light of Theorem 1 it is tempting to ask the following:

Question 1. Let Z be an absolute 1-Lipschitz retract. Is it true that for every length space X and every closed subset Y ⊆ X, any Lipschitz f : Y → Z admits an AMLE f̃: X → Z?

Note that unlike the situation when Z is a metric tree, in the setting of Question 1 one cannot expect in general that the AMLE will be unique: consider for example Z = ℓ_{∞}^{2}, i.e., the absolute 1-Lipschitz retract ℝ^{2}, equipped with the ℓ_{∞} norm. Let X = ℝ and Y = {0, 1}. The 1-Lipschitz mapping f : Y → ℓ_{∞}^{2} given by f(0) = (0, 0), f(1) = (1, 1) has many AMLEs f̃: ℝ → ℓ_{∞}^{2}, since for every 1-Lipschitz function g : ℝ → ℝ with g(0) = 0, g(1) = 1, the mapping x ↦ (x, g(x)) will be an AMLE of f. At the same time, by using the existence of real-valued AMLEs coordinate-wise, the answer to Question 1 is trivially positive when Z = ℓ_{∞}(Γ) for any set Γ.

While we do not give a general answer to Question 1, we show here that general absolute 1-Lipschitz retracts Z do enjoy a stronger Lipschitz extension property: Z-valued functions defined on subsets of vertices of 1-dimensional simplicial complexes associated to unweighted finite graphs admit isometric Lipschitz extensions which are ∞-harmonic. This issue, together with the relevant definitions, is discussed in Section 1.2 below. In addition to being crucially used in our proof of Theorem 1, this result indicates that absolute 1-Lipschitz retracts do admit enhanced Lipschitz extension theorems that go beyond the simple existence of isometric Lipschitz extensions (which is the definition of absolute 1-Lipschitz retracts). At the same time, we describe below a simple example indicating inherent difficulties in obtaining a positive answer to Question 1 beyond the class of metric trees (and their ℓ_{∞}-products).

1.2. ∞-harmonic functions and AMLEs on finite graphs. Let G = (V, E) be a finite connected (unweighted) graph. We shall consider G as a 1-dimensional simplicial complex, i.e., the edges of G are present as intervals of length 1 joining their endpoints. This makes G into a length space, where the shortest-path metric is denoted by d_{G}. Given a vertex v ∈ V denote its neighborhood in G by N_{G}(v), i.e., N_{G}(v) = {u ∈ V : uv ∈ E}.

Let (Z, d_{Z}) be a metric space. We shall say that a function f : V → Z is ∞-harmonic at v ∈ V if there exist u, w ∈ N_{G}(v) such that

d_{Z}(f(u), f(v)) = d_{Z}(f(w), f(v)) = max_{z∈N_{G}(v)} d_{Z}(f(z), f(v)),  (6)

and

d_{Z}(f(u), f(w)) = 2 max_{z∈N_{G}(v)} d_{Z}(f(z), f(v)).  (7)

(Here we allow for u = w, so that if N_{G}(v) has only one element u, then (6) and (7) imply that f(u) = f(v).) We say f : V → Z is ∞-harmonic on W ⊆ V if it is ∞-harmonic at every v ∈ W.

The connection to AMLEs is simple: for Ω ⊆ V and f : Ω → Z, if f̃ : G → Z is an AMLE of f then f̃ must be geodesic on edges, i.e., for u, v ∈ V with uv ∈ E, if x ∈ G is a point on the edge uv at distance λ ∈ [0, 1] from u, then d_{Z}(f(x), f(u)) = λd_{Z}(f(u), f(v)) and d_{Z}(f(x), f(v)) = (1 − λ)d_{Z}(f(u), f(v)) (apply (4) to the open segment joining u and v). Moreover, if G is triangle-free, then f̃ is ∞-harmonic on V ∖ Ω. This follows from considering in (4) the open set U ⊆ G consisting of the union of the half-open edges incident to v ∈ V ∖ Ω (including v itself). The vertices u, w ∈ N_{G}(v) = ∂U in (6) will be the points at which Lip_{∂U} \binom{}{f} is attained. The restriction that G is triangle-free implies that d_{G}(u, w) = 2, using which (7) follows from (4).^{1}

The converse to the above discussion is true for mappings into metric trees. This is contained in Theorem 2 below, whose simple proof appears in Section 4. A local-global statement analogous to Theorem 2 fails when the target (geodesic) metric space is not a metric tree, as we explain in Remark 1 below.

1In the above reasoning the assumption that G is triangle-free can be dropped if Z is a metric tree. But, this is not important for us: we only care about G as a length space, and therefore we can replace each edge of G by a path of length 2, resulting in a triangle-free graph whose associated 1-dimensional simplicial complex is the same as the original simplicial complex, with distances scaled by a factor of 2.

Given a metric tree T, a finite graph G = (V, E) and a function f : V → T, the linear interpolation of f is the T -valued function defined on the 1-dimensional simplicial complex associated to G as follows: given an edge e = uv ∈ E and x ∈ e with d_{G}(x, u) = λd_{G}(u, v) and d_{G}(x, v) = (1 − λ)d_{G}(u, v), the image f(x) ∈ T is the point on the geodesic joining f(u) and f(v) in T with d_{T} (f(x), f(u)) = λd_{T} (f(u), f(v)) and d_{T} (f(x), f(v)) = (1 − λ)d_{T} (f(u), f(v)).

Theorem 2. Let T be a metric tree and G = (V, E) a finite connected (unweighted) graph. Assume that Ω ⊆ V and that f : V → T is ∞-harmonic on V ∖ Ω. Then the linear interpolation of f is an AMLE of f|_{Ω}.

Remark 1. Consider the example depicted in Figure 1, viewed as a 12 vertex graph G with vertices

V = {A, B, C, X, Y, Z} ∪ {S_{i}}_{i=1}^{6}

and edges

E = {XS_{3}, S_{3}A, AS_{2}, S_{2}B, AS_{4}, S_{4}C, BS_{6}, S_{6}C, ZS_{5}, S_{5}C, Y S_{1}, S_{1}B}.

(The role of the vertices {S_{i}}_{i=1}^{6} is just to subdivide edges so that the graph will be trianglefree.) The picture in Figure 1 can also be viewed as a mapping f : V → ℝ^{2}. Denoting Ω = {X, Y, Z}, this mapping is by construction ∞-harmonic on V ∖ Ω. In spite of this fact, the linear interpolation of f is not an AMLE of f|_{Ω}. Indeed, consider the open set U = G∖Ω. Since the planar Euclidean distance between any two of the points f(X), f(Y), f(Z) is strictly less than 3 (=the distance between any two of the vertices {X, Y, Z} in G), we have Lip_{∂U}(f) = Lip_{{X,Y,Z}}(f) < 1. At the same time, by considering the vertices A, B, C we see that Lip_{U}(f) = 1.

Figure 1. An example of an ∞-harmonic function which isn’t an AMLE.

In Section 4 we show that absolute 1-Lipschitz retracts have a stronger Lipschitz extension property, namely they admit ∞-harmonic extensions for functions from finite graphs:

Theorem 3. Assume that (Z, d_{Z}) is an absolute 1-Lipschitz retract and that G = (V, E) is a finite connected (unweighted) graph. Fix Ω ⊆ V and f : Ω → Z. Then there exists a mapping f̃: V → Z which is ∞-harmonic on V ∖ Ω such that

f̃|_{Ω} = f and Lip_{V} \binom{}{f} = Lip_{Ω}(f).

The existence part of Theorem 1 is deduced in Section 5 from Theorem 3 via a compactness argument that relies on a comparison-based characterization of AMLE that we establish in Section 2. The uniqueness part of Theorem 1 is proved via a topological argument (and the results of Section 2) in Section 3.

1.3. Tug of War and Politics. In the special case when T ⊆ ℝ is an interval, Theorem 1 was proved in [42] without the local compactness assumption using a two-player, zero-sum stochastic game called Tug of War. We expect that one could adapt the arguments in [42] and the game called Politics (introduced below) to give a proof of Theorem 1 that does not use local compactness; however, this would involve rewriting large sections of [42] in a significantly more complicated way, and we will not attempt to do this here.

Tug of War is a two-player, zero-sum stochastic game. In this game, one starts with an initial point x_{0} ∈ X ∖ Y; then at the kth stage of the game, a fair coin is tossed and the winner gets to choose any x_{k} ∈ X with |x_{k} − x_{k−1}| < ε. Informally, the winning player “tugs” the game position up to ε units in a direction of her choice. The game ends the first time K that x_{K} ∈ Y, and player one collects a payoff of f(x_{K}) from player two. It was shown that as ε → 0, the value of the game (informally, the amount the first player wins in expectation when both players play optimally; see Section 6) tends to f̃(x_{0}). In addition to its usefulness in proofs, the game theory provides a deeper understanding of what an AMLE is. Although AMLEs are often difficult to compute explicitly, one can always provide upper and lower bounds by giving explicit strategies for the game and showing that they guarantee a certain expected payoff for one player or the other. It is therefore natural to ask for an analog of Tug of War that makes sense when T is not an interval.

Since f̃(x_{0}) is a point in T, however, and not in ℝ, it is not immediately obvious how f̃(x_{0}) can represent a value for either player. We will solve this problem by augmenting the state space of the game to include declared “targets” t_{k}, o_{k} ∈ T as well as “game positions” x_{k} ∈ X. Before explaining this, we remark that one obtains a slight generalization of Tug of War by letting x_{k} be vertices of any (possibly infinite) graph with vertex set X and Y ⊆ X. One then requires that x_{k} and x_{k−1} be adjacent in that graph (instead of requiring |x_{k} − x_{k−1}| < ε). We now introduce the game of Politics in a similar setting.

Let G = (V, E) be an unweighted undirected graph which may have self loops. Fix Y ⊆ V and a mapping f : Y → T. Begin with an initial game position x_{0} ∈ V ∖ Y and an initial “target” t_{0} ∈ T. At the kth round of the game, the players determine the values (x_{k}, t_{k}) as follows:

(1) Player I chooses an “opposition target” o_{k} ∈ T and collects d_{T} (o_{k}, t_{k−1}) units from player II.

(2) Player II chooses a new target t_{k} ∈ T and collects d_{T} (o_{k}, t_{k}) units from player I.

(3) A fair coin is tossed and the winner of the toss chooses a new game position x_{k} ∈ X with {x_{k−1}, x_{k}} ∈ E.

The total amount player I gains at each round is d_{T} (o_{k}, t_{k−1}) − d_{T} (o_{k}, t_{k}). Similarly, player II gains d_{T} (o_{k}, t_{k}) − d_{T} (o_{k}, t_{k−1}) at each round. The game ends after round K, where K is the smallest value of k for which x_{k} ∈ Y. At this point player I collects an additional d_{T} (f(x_{k}), t_{k}) units from player II. (If the game never ends, we declare the total payout for each player to be zero.)

The game is called “Politics” because we may view it as a model for a rather cynical zerosum political struggle in which f(x_{K}) represents a “political outcome,” but both parties care only about their own perceived political strength, and not about the actual outcome. We think of the target as representing the “declared political objective” of player II; the terminal payoff rule, makes it clear that player II would prefer f(x_{K}) be close to this declared target (in order to “appear successful”). Player II is allowed to adjust the target during each round, but loses points for moving her target closer to the declared opposition target o_{k} (because “making a concession” makes her appear weak) and gains points for moving her target further from the opposition target because “taking a harder line” makes her appear strong). ^{2}

We will prove the following for finite graphs:

Proposition 4. Fix a finite graph G = (V, E), some Y ⊆ V, a metric tree T, and a function f : Y → T. View G as a length space (with all edges having length one) and let f̃: G → T be the AMLE of f. Then the value of the game of Politics with these parameters and initial vertex x_{0} ∈ V ∖ Y is given by

d_{T} (f̃(x_{0}), t_{0}) .

Proposition 4 will be proved in Section 6. An extension of Proposition 4 to infinite graphs (via the methods of [42]) is probably possible, but we will not attempt it here.

1.4. Some open questions and directions for future research. It would be of interest to understand known isometric extension theorems in the context of the AMLE problem. Specifically, we ask:

Question 2. Is there an AMLE version of Kirszbraun’s extension theorem, i.e, is it true that for every pair of Hilbert spaces H_{1}, H_{2} and every closed subset Y ⊆ H_{1}, any Lipschitz mapping f : Y → H_{2} admits an AMLE f̃: H_{1} → H_{2}?

We refer to the manuscript [44] for a discussion of subtleties related to Question 2, as well as some partial results in this direction. Examples of additional isometric extension theorems that might have AMLE versions are contained in [45, 46, 47, 27, 39].

The study of isomorphic extensions in the context of the AMLE problem is wide open. Since when Lipschitz extension is possible a constant factor loss is usually necessary, and since isomorphic extensions suffice for many applications, it would be of interest if some isomorphic extension theorems had “almost locally optimal” counterparts. For example, one might ask for the existence of a constant K > 0 such that one can extend any mapping f : Y → Z to a mapping f̃: X → Z so that for every open U ⊆ X ∖ Y we have

2There is a more player-symmetric variant of this game in which each player, upon moving a target, earns the net change in the distance from the opponent’s target. That is, player II earns d_{T} (o_{k}, t_{k}) − d_{T} (o_{k}, t_{k−1})

when choosing t_{k} (so player I earns d_{T} (o_{k}, t_{k−1})−d_{T} (o_{k}, t_{k})) and player I earns d_{T} (o_{k}, t_{k−1})−d_{T} (o_{k−1}, t_{k−1})

when choosing o_{k}. In fact, by combining like terms, modifying the end-of-game payout function, and defining o_{0} = t_{0}, one can make this game equivalent to the one described above but with twice the total payout.

Lip_{U} \binom{}{f} ⩽ K · Lip_{∂U} \binom{}{f} .  (8)

Examples of isomorphic extension results that could be studied in the context of the AMLE problem include [31, 32, 18, 20, 5, 43, 25, 9, 30, 40, 8, 36, 22, 26]. Unlike isometric extension theorems, isomorphic extension theorems cannot be done “one point at time”, since naïvely the constant factor losses at each step would accumulate. For this reason, isomorphic extension theorems usually require methods that are very different from their isometric counterparts. One would therefore expect that entirely new approaches are necessary in order to prove AMLE versions of isomorphic extension.

1.5. Possible applications. The image processing literature makes use of real-valued AM-LEs as a technique for image inpainting and surface reconstruction — see [11, 1, 35, 10]. Since many data sets in areas ranging from computer science to biology have a natural tree structure, it stands to reason that problems involving reconstruction/interpolation of missing tree-valued data could be similarly approached using tree valued AMLEs.

Tree-valued AMLEs may also be useful for problems that do not involve trees a priori. To give a simple illustration of this, suppose we have a two-dimensional surface S embedded in ℝ^{3} that separates an “inside” from an “outside,” but such that on some open W ⊆ ℝ^{3} the shape of the surface is not known. Let d(x) be the signed distance of x from S (i.e., the actual distance if x is on the outside and minus that distance if x is on the inside). If we can compute or approximate d(x) outside of W, then the extension of d(x) to W has a zero set that can be interpreted as a “reconstructed” approximation to S. This approach and related methods are explored in [10].

If instead of a single “inside” and “outside” there were three or more regions of space meeting at a point v, and the union S of the interfaces between these regions was unknown in a neighborhood W of v, then we could use the same approach but replace ℝ with the metric tree ⋃ ω_{i}[0, ∞) ⊆ ℂ for some complex roots of unity ω_{i}, and let d(x) be ω_{i} (when x is in the ith region) times the distance from x to S. A similar technique could be used for inpainting a two-dimensional image comprised of a small number of monochromatic regions. Indeed, for such problems, it is not clear how one could apply the AMLE method without using trees.

2. Comparison formulation of absolute minimality

We take the following definition from [12] (see also [17, 13] for the case X = ℝ^{n}). Let U be an open subset of a length-space (X, d_{X}) and let f : \overline{U} → ℝ be continuous. Then f is said to satisfy comparison with distance functions from above on U if for every open W ⊆ U, z ∈ X ∖ W, b ⩾ 0 and c ∈ ℝ we have the following:

(∀ x ∈ ∂W f(x) ⩽ b d_{X}(x, z) + c) ⟹ (∀ x ∈ W f(x) ⩽ b d_{X}(x, z) + c).  (9)

The function f is said to satisfy comparison with distance functions from below on U if the function −f satisfies comparison with distance functions from above on U, i.e., for every open W ⊆ U, z ∈ X ∖ W, b ⩾ 0 and c ∈ ℝ we have the following:

(∀ x ∈ ∂W f(x) ⩾ −b d_{X}(x, z) + c) ⟹ (∀ x ∈ W f(x) ⩾ −b d_{X}(x, z) + c). (10)

Finally, f satisfies comparison with distance functions on U if it satisfies comparison with distance functions from above and from below on U. We cite the following:

Proposition 5 ([12]). Let U be an open subset of a length space. A continuous f : \overline{U} → ℝ satisfies comparison with distance functions on U if and only if it is an AMLE of f|_{∂U}.

Remark 2. The definition of comparison with distance functions from above would not change if we added the requirement that z ∉ ∂W; if (9) or (10) fails and z ∈ ∂W, then it will fail (with a modified c) when W is modified to include some neighborhood of z. The definition would also not change if we required b > 0. If (9) or (10) fails with b = 0, then it fails for some sufficiently small b′ > 0.

We will need to have an analog of the above definition with the real line ℝ replaced with T. The definition makes sense when T is any metric space, but we will only use it in the case when T is a metric tree. We say f : \overline{U} → T satisfies T -comparison on U if for every t ∈ T, the function x ↦ d_{T} (t, f(x)) satisfies comparison with distance functions from above on U. This generalizes comparison with distance functions:

Proposition 6. If T is the closed interval [t_{1}, t_{2}] ⊆ ℝ, then f : \overline{U} → T satisfies T -comparison on U if and only if it satisfies comparison with distance functions on U.

Proof. If f satisfies T -comparison on U, then the mappings x ↦ d_{T} (t_{1}, f(x)) = f(x) − t_{1} and x ↦ d_{T} (t_{2}, f(x)) = t_{2} − f(x) satisfy comparison with distance functions from above on U, hence f and −f both satisfy comparison with distance functions from above. Conversely, if f satisfies comparison with distance functions on U, then for all t ∈ [t_{1}, t_{2}] the mapping x ↦ d_{T} (f(x), t) = (f(x) − t) ∨ (t − f(x)) satisfies comparison with distance functions from above because it is a maximum of two functions with this property. □

Proposition 5 also has a natural generalization, which is contained in Proposition 7 below. Note that the proof of this generalization uses the assumption that T is a metric tree in the “only if” direction; for the “if” direction T can be any metric space.

Proposition 7. Let U be an open subset of a length space (X, d_{X}), and let (T, d_{T}) be a metric tree. A continuous function f : \overline{U} → T satisfies T -comparison on U if and only if it is an AMLE of f|_{∂U}.

Proof. We will first suppose, to obtain a contradiction, that f is not an AMLE of f|_{∂U}, but satisfies T -comparison. Then there is an open W ⊆ U such that Lip_{W} (f) > Lip_{∂W} (f). That is, there is a path P in X connecting points x and y in W whose length L satisfies

\frac{d_{T} (f(x), f(y))}{L} > Lip_{∂W} (f).  (11)

If y_{1} and y_{2} are the first and last times P hits ∂W, d_{T} (f(y_{1}), f(y_{2})) ⩽ Lip_{∂W} (f) · d_{X}(y_{1}, y_{2}); hence the property (11) holds for either the portion of P between x and y_{1} or the portion between y_{2} and y. Thus, we may take P to be entirely contained in W; replacing P with a slightly shorter sub-path of P, we may assume the endpoints of P are both in W as well, and that P is some positive distance δ from ∂W. Set

m =^{def} \frac{d_{T} (f(x), f(y))}{L} > Lip_{∂W} (f).

We may then find x_{1} arbitrarily close to some fixed point x_{0} along P satisfying

\frac{d_{T} (f(x_{0}), f(x_{1}))}{d_{X}(x_{0}, x_{1})} ⩾ m > Lip_{∂W} (f).

Now we consider the distance function md_{X}(x_{0}, ·). We will compare it to the function d_{T} (f(x_{0}), f(·)). Since the latter is at least as large as the former at the point x_{1}, T -comparison implies that it must be at least as large at some point on ∂W. This implies that for any ε > 0 we may find a z ∈ ∂W where

m′ =^{def} \frac{d_{T} (f(x_{0}), f(z))}{d_{X}(x_{0}, z)} ⩾ m − ε.

In particular, we may assume m′ > Lip_{∂W} (f). Next choose m′′ ∈ (Lip_{∂W} (f), m′). Consider the distance function m′′d_{X}(z, ·) and compare it to d_{T} (f(z), f(·)). Since the functions are equal at z and the latter is larger than the former at x_{0}, the latter must be larger than the former at some point w ∈ (∂W) ∖ {z}. But this implies

\frac{d_{T} (f(z), f(w))}{d_{X}(z, w)} > Lip_{∂W} (f),

a contradiction.

We now proceed to the converse. Note that since T is a bounded metric space, by intersecting U with a large ball it suffices to prove the converse when U is bounded. Suppose, to obtain a contradiction, that f is an AMLE of f|_{∂U} and does not satisfy T -comparison on U. Since f does not satisfy T -comparison on U, there exists an open W ⊆ U, a point x_{0} ∉ W and c ∈ ℝ, b ⩾ 0, such that for some t ∈ T we have d_{T} (t, f(x)) ⩽ bd_{X}(x_{0}, x) + c for all x ∈ ∂W, yet d_{T} (t, f(y)) > bd_{X}(x_{0}, y) + c for some y ∈ W. Write F (z) = bd_{X}(x_{0}, z) + c. We may replace W with the connected component of {x ∈ W : d_{T} (t, f(x)) > F (x)} containing y, so that one has d_{T} (t, f(x)) = F (x) at the boundary of W. By looking at a nearly-shortest path from y to x_{0}, we deduce that Lip_{W} (f) > b. If we could also show that Lip_{∂W} (f) = b (which is trivially the case when T ⊆ ℝ, but not for a more general metric tree T) we would have a contradiction to the AMLE property of f. Instead of proving this for the particular W constructed above, we will show that there exists a smaller W for which the analogous statement holds.

Consider the function

G(s) =^{def} sup_{\begin{matrix}x∈\overline{W} \\ d_{X}(x_{0},x)=s \end{matrix}} d_{T} (t, f(x)),

which is defined on the interval (s_{1}, s_{2}), where s_{1} and s_{2} are the infimum and supremum of the set {d_{X}(x_{0}, x) : x ∈ W }, respectively. By assumption G(s) lies above the line bs + c for some s ∈ (s_{1}, s_{2}), though not for s_{1} and s_{2}. Hence, if we define

M =^{def} sup {G(s) − bs − c : s ∈ (s_{1}, s_{2})}

then M > 0. Write

S =^{def} \{σ ∈ [s_{1}, s_{2}] : lim sup_{\begin{matrix}s→σ \\ s∈(s_{1},s_{2}) \end{matrix}} (G(s) − bs − c) = M\} ,

and note that S is a nonempty closed subset of [s_{1}, s_{2}], so that s_{0} =^{def} inf S ∈ S. For ε > 0 and x ∈ X define

F_{ε}(x) =^{def} (b + ε)d_{X}(x_{0}, x) + M + c − εs_{0} − ε^{2},

and

W_{ε} =^{def} {x ∈ W : d_{T} (t, f(x)) > F_{ε}(x)}.

Observe that W_{ε} ≠ ∅ for all ε > 0. To see this fix δ > 0. Since s_{0} ∈ S there exists s ∈ (s_{1}, s_{2}) such that |s − s_{0}| ⩽ δ and G(s) − bs − c ⩾ M − δ. By the definition of G(s), there is z_{0} ∈ \overline{W} satisfying d_{X}(z_{0}, x_{0}) = s and G(s) ⩽ d_{T} (t, f(z_{0})) + δ. Since f is continuous at z_{0}, there is η ∈ (0, δ) such that if d_{X}(z, z_{0}) < η then d_{T} (f(z), f(z_{0})) < δ. Take z ∈ W with d_{X}(z, z_{0}) < η. Then,

d_{T} (t, f(z)) > d_{T} (t, f(z_{0})) − δ
⩾ G(s) − 2δ
⩾ M + bs + c − 3δ
⩾ M + bs_{0} + c − (3 + b)δ
= F_{ε}(z) − (b + ε)d_{X}(x_{0}, z) + bs_{0} + εs_{0} + ε^{2} − (3 + b)δ
> F_{ε}(z) − (b + ε)(s + η) + bs_{0} + εs_{0} + ε^{2} − (3 + b)δ
⩾ F_{ε}(z) − (b + ε)(s_{0} + δ + η) + bs_{0} + εs_{0} + ε^{2} − (3 + b)δ
> F_{ε}(z) + ε^{2} − (3δ + 2εδ + 3bδ).

Thus for δ small enough we have z ∈ W_{ε}. The following claim contains additional properties of the sets W_{ε} that we will use later.

Claim 8. The open sets {W_{ε}}_{ε>0} have the following properties:

(1) If 0 < ε_{1} < ε_{2} then \overline{W_{ε_{1}}} ⊆ W_{ε_{2}},

(2) lim_{ε→0} sup_{x∈W_{ε}} |d_{X}(x_{0}, x) − s_{0}| = 0,

(3) lim_{ε→0} sup_{x∈W_{ε}} |d_{T} (t, f(x)) − (M + bs_{0} + c)| = 0.

Proof. Fix 0 < ε_{1} < ε_{2} and x ∈ \overline{W_{ε_{1}}}. Write s = d_{X}(x_{0}, x). Since d_{T} (t, f(x)) ⩾ F_{ε_{1}}(x), we have G(s)−bs−c ⩾ ε_{1}s+M −ε_{1}s_{0} −ε_{1}^{2}. By the definition of M, this implies that s ⩽ s_{0} +ε_{1}. Hence,

d_{T} (t, f(x)) ⩾ (b + ε_{1})s + M + c − ε_{1}s_{0} − ε_{1}^{2} = F_{ε_{2}}(x) + (ε_{2} − ε_{1})s_{0} + ε_{2}^{2} − ε_{1}^{2} − (ε_{2} − ε_{1})s

⩾ F_{ε_{2}}(x) + (ε_{2} − ε_{1})s_{0} + ε_{2}^{2} − ε_{1}^{2} − (ε_{2} − ε_{1})(s_{0} + ε_{1}) = F_{ε_{2}}(x) + ε_{2}(ε_{2} − ε_{1}) > F_{ε_{2}}(x).

Thus x ∈ W_{ε_{2}}, proving the first assertion of Claim 8.

To prove the second assertion of Claim 8, note that we have already proved above that if x ∈ W_{ε} then d_{X}(x_{0}, x) ⩽ s_{0} + ε. Thus, if the second assertion of Claim 8 fails there is some δ > 0 and a sequence {ε_{n}}_{n=1}^{∞} ⊆ [0, 1] with lim_{n→∞} ε_{n} = 0, such that for each n ∈ ℕ there is z_{n} ∈ W_{ε_{n}} with d_{X}(z_{n}, x_{0}) ⩽ s_{0} − δ. Write σ_{n} = d_{X}(z_{n}, x_{0}), and by passing to a subsequence assume that lim_{n→∞} σ_{n} = σ_{∞} exists. Then σ_{∞} ⩽ s_{0} − δ and,

lim sup_{n→∞} (G(σ_{n}) − bσ_{n} − c) ⩾ lim sup_{n→∞} d_{T} (t, f(z_{n})) − bσ_{∞} − c ⩾ lim sup_{n→∞} F_{ε_{n}}(z_{n}) − bσ_{∞} − c

= lim sup_{n→∞} ((b + ε_{n})σ_{n} + M + c − ε_{n}s_{0} − ε_{n}^{2}) − bσ_{∞} − c = M.

Thus σ_{∞} ∈ S. But since σ_{∞} ⩽ s_{0} − δ, this contradicts the choice of s_{0} as the minimum of S. The proof of the second assertion of Claim 8 is complete. The third assertion of Claim 8 now follows, since if x ∈ W_{ε} then by writing s = d_{X}(x, x_{0}) we see that

b(s − s_{0}) ⩾ [(G(s) − bs − c) − M] + b(s − s_{0}) ⩾ d_{T} (t, f(x)) − (M + bs_{0} + c)

⩾ F_{ε}(x) − (M + bs_{0} + c) = b(s − s_{0}) + εs − εs_{0} − ε^{2}.

Thus

sup_{x∈W_{ε}} |d_{T} (t, f(x)) − (M + bs_{0} + c)| ⩽ b sup_{x∈W_{ε}} |d_{X}(x_{0}, x) − s_{0}| + εs_{0} + ε^{2},

and therefore the third assertion of Claim 8 follows from the second assertion of Claim 8. □

We are now in position to conclude the proof of Proposition 7. Let V be the the set of vertices of the metric tree T. We claim that for all ε > 0 such that εs_{0} + ε^{2} ⩽ M we have f(W_{ε}) ∩ V ≠ ∅ (recall that by our assumption we have M > 0). Indeed, if x ∈ ∂W_{ε} then either d_{T} (t, f(x)) = F_{ε}(x) or x ∈ ∂W. In the latter case, by assumption we have d_{T} (t, f(x)) ⩽ bd_{X}(x_{0}, x) + c ⩽ F_{ε}(x), where the last inequality follows from εs_{0} + ε^{2} ⩽ M. Thus, by the definition of W_{ε}, the function f does not satisfy T -comparison on W_{ε}. Since f is an AMLE of f|_{∂U}, Proposition 5, combined with Proposition 6, now implies that f|_{W_{ε}} must take values in V.

Due to part (1) of Claim 8, there exists v ∈ V such that v ∈ ⋂_{ε>0} W_{ε}. Let W′_{ε} be the connected component of W_{ε} whose image under f contains v. By part (3) of Claim 8, for ε small enough we have f(W′_{ε}) ∩ V = {v}. Since, by the definition of W_{ε} and the connectedness of W′_{ε}, for x ∈ ∂W′_{ε} we have d_{T} (t, f(x)) = F_{ε}(x), by considering a nearlyshortest path from a point in W′_{ε} to x_{0} we see that Lip_{W′_{ε}}(f) > b + ε. Since f is an AMLE of f|_{∂U}, it follows that Lip_{∂W′_{ε}}(f) > b + ε. This implies that there are distinct x_{ε}, y_{ε} ∈ ∂W′_{ε} such that d_{T} (f(x_{ε}), f(y_{ε})) > (b + ε)d_{X}(x_{ε}, y_{ε}). But, since d_{T} (t, f(x_{ε})) = F_{ε}(x_{ε}) and d_{T} (t, f(y_{ε})) = F_{ε}(y_{ε}), it must be the case that the distance from t of both f(x_{ε}) and f(y_{ε}) is at least their distance from v. Indeed, if t ∈ ⋂_{ε>0} f(W′_{ε}) then it would follow from part (3) of Claim 8 that t = v, and there is nothing to prove. Otherwise, for ε small enough t ∉ f(W′_{ε}), and therefore, since T is a tree, if at least one of the points f(x_{ε}), f(y_{ε}) is closer to t than to v then the points t, f(x_{ε}), f(y_{ε}) all lie on the same geodesic in T, implying that:

d_{T} (f(x_{ε}), f(y_{ε})) = |d_{T} (t, f(x_{ε})) − d_{T} (t, f(y_{ε}))| = |F_{ε}(x_{ε}) − F_{ε}(y_{ε})|

= (b + ε)|d_{X}(x_{0}, x_{ε}) − d_{X}(x_{0}, y_{ε})| ⩽ (b + ε)d_{X}(x_{ε}, y_{ε}),

a contradiction to the choice of x_{ε}, y_{ε}.

Having proved that both f(x_{ε}) and f(y_{ε}) lie further away from t than v, if we consider a nearly shortest-path between x_{ε} and y_{ε}, it must include points x_{1}, x_{2} ∈ \overline{W′_{ε}} such that f(x_{1}) and f(x_{2}) lie on the same component I of T ∖ V — on the other side of v from t — and satisfy

\frac{d_{T} (f(x_{1}), f(x_{2}))}{d_{X}(x_{1}, x_{2})} > (b + ε).  (12)

Suppose that f(x_{1}) is closer to t than f(x_{2}). Note that

d_{T} (f(x_{1}), f(x_{ε})) = d_{T} (t, f(x_{ε})) − d_{T} (t, f(x_{1})) ⩽ F_{ε}(x_{ε}) − F_{ε}(x_{1})

= (b + ε)(d_{X}(x_{0}, x_{ε}) − d_{X}(x_{0}, x_{1})) ⩽ (b + ε)d_{X}(x_{ε}, x_{1}).

Moreover, if x = x_{1} then trivially d_{T} (f(x_{1}), f(x)) ⩽ (b + ε)d_{X}(x_{1}, x). Thus, if we let J ⊆ I be the open interval joining f(x_{ε}) and f(x_{1}), then d_{T} (f(x_{1}), f(·)) ⩽ (b + ε)d_{X}(x_{1}, ·) on ∂ (f^{−1}(J) ∩ W′_{ε}). By (12) we now have a violation of T -comparison on f^{−1}(J), which contradicts Proposition 5. □

3. Uniqueness

In this section we prove the uniqueness half of Theorem 1 (which does not require the locally compact assumption) as Lemma 12 below. Before doing so, we prove some preliminary lemmas.

Lemma 9. Suppose that X is a length space, that Y ⊆ X is closed, that f : Y → ℝ is Lipschitz and bounded, and that f̃: X → ℝ is the AMLE of f. (Existence and uniqueness of f̃ are proved in [42].) Suppose that g : X → ℝ is another bounded and continuous extension of f, and that for some fixed δ > 0, this g satisfies comparison with distance functions from above on every radius δ ball centered in X ∖ Y. Then g ⩽ f̃ on X.

Proof. This is proved (though not explicitly stated) in [42]. Precisely, it is shown there that for ε > 0, the comparison with distance functions from above on balls of radius larger than 2ε implies that the first player in a modified tug of war game (with game position v_{k} and step size ε) can make g(v_{k}) a submartingale until the termination of the game, which in turn implies that g ⩽ f_{ε} where f_{ε} is the value of this game. It is also shown that lim_{ε→0} f_{ε} = f̃ holds on X. Taking ε → 0 (and noting 2ε < δ for small enough ε) gives g ⩽ f̃. □

The following was proved in [2, Lem. 5]. The statement in [2] was only made for the special case X ⊆ ℝ^{n}, but the (short) proof was not specific to ℝ^{n}. For completeness, we copy the proof from [2], adapted to our notation. We will vary the presentation just slightly — using suprema over open balls instead of maxima over closed balls — because in our context (since we do not assume any kind of local compactness) maxima of continuous functions on closed balls are not necessarily obtained.

Lemma 10. Let (X, d_{X}) be a length space, x_{0} ∈ X and ε > 0. Suppose that f : X → ℝ satisfies comparison with distance functions from above on a domain containing B(x_{0}, 2ε). Write

f^{ε}(x) =^{def} sup_{B(x,ε)} f, f_{ε}(x) =^{def} inf_{B(x,ε)} f

and

S_{ε}^{+}f(x) =^{def} sup_{y∈B(x,ε)} \frac{f(y) − f(x)}{ε}, S_{ε}^{−}f(x) =^{def} sup_{y∈B(x,ε)} \frac{f(x) − f(y)}{ε}.

Then

S_{ε}^{−}f^{ε}(x_{0}) ⩽ S_{ε}^{+}f^{ε}(x_{0}).

Proof. For δ > 0 we may select y_{0} ∈ B(x_{0}, ε) and z_{0} ∈ B(x_{0}, 2ε) such that |f(y_{0})−f^{ε}(x_{0})| ⩽ δ and |f(z_{0}) − f^{2ε}(x_{0})| ⩽ δ. Then,

ε (S_{ε}^{−}f^{ε}(x_{0}) − S_{ε}^{+}f^{ε}(x_{0})) = 2f^{ε}(x_{0}) − (f^{ε})^{ε}(x_{0}) − (f^{ε})_{ε}(x_{0})
⩽ 2f^{ε}(x_{0}) − f^{2ε}(x_{0}) − f(x_{0})
⩽ 2f(y_{0}) − f(z_{0}) − f(x_{0}) + 2δ,  (13)

where we used the fact that (f^{ε})^{ε}(x_{0}) = f^{2ε}(x_{0}) (since X is a length space), and that by definition (f^{ε})_{ε}(x_{0}) ⩾ f(x_{0}).

Note that if d_{X}(w, x_{0}) = 2ε then

f(w) ⩽ f^{2ε}(x_{0}) = f(x_{0}) + \frac{f^{2ε}(x_{0}) − f(x_{0})}{2ε}d_{X}(w, x_{0}).

Hence for all w ∈ ∂(B(x_{0}, 2ε) ∖ {x_{0}}) we have

f(w) ⩽ f(x_{0}) + \frac{f^{2ε}(x_{0}) − f(x_{0})}{2ε}d_{X}(w, x_{0}).  (14)

Since f^{2ε}(x_{0}) − f(x_{0}) ⩾ 0, we may apply the fact that f satisfies comparison with distance functions from above to deduce that (14) holds for every w ∈ B(x_{0}, 2ε) ∖ {x_{0}}, and thus for every w ∈ B(x_{0}, 2ε). Substituting w = y_{0}, we see that

2f(y_{0}) − f(x_{0}) − f(z_{0}) ⩽ f(x_{0}) − f(z_{0}) + \frac{f^{2ε}(x_{0}) − f(x_{0})}{ε}d_{X}(y_{0}, x_{0})
⩽ − (1 − \frac{d_{X}(y_{0}, x_{0})}{ε}) (f^{2ε}(x_{0}) − f(x_{0})) + δ
⩽ δ,  (15)

where we used the fact that d_{X}(y_{0}, x_{0}) ⩽ ε. Since (15) holds for all δ > 0, the required result follows from a combination of (13) and (15). □

Lemma 11. Assume the following structures and definitions:

(1) A length space X, a closed Y ⊆ X, a metric tree T, and a Lipschitz f : Y → T.

(2) A fixed x_{0} ∈ X ∖ Y and the set X̂ defined as the space of finite-length closed paths in X (parameterized at unit speed) that begin at x_{0} and remain in X ∖ Y except possibly at right endpoints.

(3) A metric on X̂ defined as follows: given û, v̂ ∈ X̂, dX̂ (û, v̂) is the sum of the lengths of the portions of the two paths that occur after the largest time at which û and v̂ agree. (Note that X̂ is an ℝ-tree under this metric.)

(4) The “covering map” M : X̂ → X that sends a path in X̂ to its right endpoint.

If f̃: X → T is an AMLE of f, then f̂ =^{def} f̃∘ M : X̂ → T is an AMLE of f ∘ M (which is defined on Ŷ =^{def} M^{−1}(Y)).

Proof. First we claim that M is path-length preserving. That is, if γ̂ is any rectifiable path in X̂ then γ =^{def} M ∘ γ̂ is a rectifiable path of the same length in X. This is true by definition if L ∘ γ̂ (here L(·) denotes path length) is strictly increasing, and similarly if L ∘ γ̂ is strictly decreasing. Since the length of γ̂ is the total variation of L ∘ γ̂ (and the latter is finite), the general statement can be derived by approximating γ̂ with paths for which L ∘ γ̂ is piecewise monotone. To do this, first note that one can take an increasing set of times 0 = t_{0}, t_{1}, …, t_{k} such that the total variation of L ∘ γ̂ restricted to those times is arbitrarily close to the unrestricted total variation. Then the length of γ traversed between times t_{j} and t_{j+1} is at least r =^{def} |L ∘ γ̂(t_{j}) − L ∘ γ̂(t_{j+1})| — this is because the longer of the two paths γ̂(t_{j}) and γ̂(t_{j+1}) contains a segment of length at least r that is not part of the other, and γ must traverse all the points of that segment in order (or in reverse order) somewhere between times t_{j} and t_{j+1}.

Consider an open subset Ŵ ⊆ X̂ ∖ Ŷ and note that W =^{def} M \binom{}{W} is also open. We need

to show that if f̃ is an AMLE of f then we cannot have LipŴ \binom{}{f} > Lip_{∂Ŵ} \binom{}{f}.

Indeed, suppose we had LipŴ \binom{}{f} > Lip_{∂Ŵ} \binom{}{f}. Then we could find a path γ̂ within Ŵ

connecting points a, b ∈ Ŵ such that

\frac{dX̂ (f̂(a), f̂(b))}{L(γ)} > m

for some

m > Lip_{∂Ŵ} \binom{}{f} .

Since dX̂ (f̂(a), f̂(γ̂(s))) is Lipschitz (hence a.e. differentiable) in s, we can find an s_{0} at which its derivative is greater than m. Thus, for all sufficiently small ε_{0}, writing x_{0} = γ(s_{0}) (where γ = M ∘ γ̂), we can find points x_{1}, x_{−1} ∈ W such that d_{X}(x_{1}, x_{0}) = d_{X}(x_{−1}, x_{0}) = ε_{0} and f̃(x_{1}), f̃(x_{−1}) are both at distance greater than mε_{0} from f̃(x_{0}) and lie in distinct components of T ∖ {f̃(x_{0})}.

Now consider some ε > 0 much smaller than ε_{0}. Since f̃ is an AMLE of f, the T - comparison property implies that the function

g(·) =^{def} d_{T} (f̃(·), f̃(x_{0}))

satisfies comparison with distance functions from above. Moreover, along any near-geodesic from x_{0} to x_{1}, the function g increases at an average speed greater than m. Write, as before, g^{ε}(x) =^{def} sup_{B(x,ε)} g. Let C be the Lipschitz constant of f̃, so that |g − g^{ε}| ⩽ Cε. Note that g(x_{0}) = 0 and g(x_{1}) > mε_{0}, and therefore g^{ε}(x_{1}) − g^{ε}(x_{0}) > mε_{0} − 2Cε. By considering a near-geodesic from x_{0} to x_{1}, this implies that when ε is small enough, we can find points y_{1}, y_{2} ∈ B(x_{0}, ε_{0}) with g^{ε}(y_{2}) − g^{ε}(y_{1}) > mε and d_{T} (y_{1}, y_{2}) ⩽ ε, such that f̃(y_{1}) and f̃(y_{2}) lie in the same component of T ∖ {f̃(x_{0})} as f̃(x_{1}), and both f̃(y_{1}) and f̃(y_{2}) are at distance

at least 3Cε from f̃(x_{0}).

Fix δ > 0. Applying Lemma 10 inductively we obtain a sequence of points {y_{i}}_{i=1}^{k} such that for all i ∈ {1, …, k − 1} we have d_{X}(y_{i}, y_{i+1}) ⩽ ε and for all i ∈ {1, …, k − 2},

g^{ε}(y_{i+2}) − g^{ε}(y_{i+1}) ⩾ g^{ε}(y_{i+1}) − g^{ε}(y_{i}) − \frac{δ}{2^{i}} .  (16)

This iterative construction can continue until the first k for which y_{k} has distance at most ε from ∂W. It follows from (16) that

g^{ε}(y_{i+1}) − g^{ε}(y_{i}) ⩾ g^{ε}(y_{2}) − g^{ε}(y_{1}) − ∑_{j=1}^{∞} \frac{δ}{2^{j}} > mε − δ.

Thus, assuming δ is small enough, we have g^{ε}(y_{i+1}) − g^{ε}(y_{i}) > mε for all i ⩾ 1. It follows that for all j > i ⩾ 1 we have

g^{ε}(y_{j}) > g^{ε}(y_{i}) + (j − i)mε.  (17)

A consequence of (17) is that, since g^{ε} is bounded, the above construction cannot continue indefinitely without reaching a point within ε distance from ∂W.

Another consequence of (17) and the fact that f̃ is C-Lipschitz is that for all j > i,

g(y_{j}) > g(y_{i}) + (j − i)mε − 2Cε.  (18)

In particular, since g(y_{1}) ⩾ 3Cε, it follows from (18) that for all i ⩾ 1 we have g(y_{i}) ⩾ Cε. This implies that the points {f̃(y_{i})}_{i=1}^{∞} are all in the same component of T ∖ {f̃(x_{0})} as

f̃(x_{1}), since otherwise if i is the first index such that f̃(y_{i}) is not in this component, then d_{T} (f̃(y_{i}), f̃(y_{i−1})) ⩾ 2Cε, contradicting the fact that f̃ is C-Lipschitz.

We similarly construct the sequence {z_{i}}_{i=1}^{ℓ}, starting with the point x_{−1} instead of the point x_{1}, such that for all j > i,

g(z_{j}) > g(z_{i}) + (j − i)mε − 2Cε.

As before, the entire sequence {f̃(z_{i})}_{i=1}^{ℓ} must remain in the same component of T ∖{f̃(x_{0})} as f(x_{−1}) and z_{ℓ} is within ε distance from ∂W.

Now consider some M pre-image x̂_{0} of x_{0} that is contained in Ŵ, and let D denote the the distance from x̂_{0} to ∂Ŵ. Among the rectifiable paths in W from one boundary point of W to another that pass through all the z_{k} in reverse order and the subsequently the y_{k} in order, let γ_{0} be one which is near the shortest. Take any path γ̂_{0} through x̂_{0} such that M ∘ γ̂_{0} = γ_{0}. Then consider a maximal arc of this path contained in Ŵ and containing x̂_{0} (which necessarily has length at least D and connects two points on ∂Ŵ). The length of this maximal arc is at least D and the change in f̂ from one endpoint of the arc to the other is (in distance) at least m times the length of the arc, plus an O(ε_{0}) error, which contradicts the definition of m. □

Lemma 12. Given a length space (X, d_{X}), a closed Y ⊆ X, a metric tree (T, d_{T}), and a Lipschitz function f : Y → T, there exists at most one f̃: X → T which is an AMLE of f.

Proof. By Proposition 7 we must show that if g, h : X → T are continuous functions such that g = h on Y, and g, h both satisfy T -comparison on X ∖ Y, then g = h throughout X. By Lemma 11, it is enough to prove this in the case that X is an ℝ-tree: in particular, we may assume that X and X ∖ Y are simply connected (note that the connected components of an open subset of a metric tree are simply connected).

For the sake of obtaining a contradiction, suppose that g, h satisfy these hypotheses, but g(x) ≠ h(x) for some x ∈ X ∖ Y. Then the hypotheses still hold if we replace Y with complement of the connected component of {y ∈ X : g(y) ≠ h(y)} containing x, and redefine g to be equal to h on all other connected components of {y ∈ X : g(y) ≠ h(y)}. In other words, we lose no generality in assuming g(x) ≠ h(x) for all x ∈ X ∖ Y. The pair (g(·), h(·)) may then be viewed as a map from X ∖ Y to

T =^{def} {(t_{1}, t_{2}) ∈ T × T : t_{1} ≠ t_{2}}.

We will use this map to define a certain pair of real-valued functions on X.

To this end, consider an arbitrary continuous function P : [0, 1] → T . Define functions t_{1}, t_{2} : [0, 1] → T by writing (t_{1}(s), t_{2}(s)) = P (s). Let I((t_{1}, t_{2})) denote the geodesic joining t_{1} and t_{2} in T. For every s ∈ [0, 1], we will define an isometry Ψ_{s}^{P} from I(P (s)) to an interval (a_{1}(s), a_{2}(s)) of ℝ, sending t_{1} to a_{1}(s) and t_{2} to a_{2}(s). Clearly, the values of a_{1}(s) and a_{2}(s) determine the isometry, and for each s, we must have

a_{2}(s) − a_{1}(s) = d_{T} (t_{1}(s), t_{2}(s)).

However, the latter observation only determines a_{1}(s) and a_{2}(s) up to the addition of a single constant to both values. This constant is determined by the following requirements:

(1) a_{1}(0) = 0.

(2) For each fixed t ∈ T, the function s ↦ Ψ_{s}^{P} (t) is constant on every connected interval

of the (open) set {s ∈ [0, 1] : t ∈ I(P (s))}. Informally, at each time s, the geodesic I(P (s)) is “glued” isometrically to the interval (a_{1}(s), a_{2}(s)) ⊆ ℝ. As s increases, if t_{1} and t_{2} move closer to each other, then points are being removed from the ends of I(P (s)), and these points are “unglued” from ℝ. As t_{1} and t_{2} move further from each other, new points are added to the geodesic and these new points are glued back onto ℝ.

When P′ and P are paths in T as above, write P ∼ P′ if P (0) = P′(0) and P (1) = P′(1) and for each s ∈ [0, 1] we have I(P (s)) ∩ I(P′(s)) ≠ ∅. We claim that in this case Ψ_{1}^{P} = Ψ_{1}^{P′}. Note that given s and t ∈ I(P (s)) ∩ I(P′(s)), requirement (2) above implies that Ψ_{s}^{P} and Ψ_{s}^{P′} agree up to additive constant on a neighborhood of s; namely, they must agree on the connected component of {s′ : t ∈ I(P (s′))∩I(P′(s′))} containing s. Since Ψ_{s}^{P} and Ψ_{s}^{P′} agree up to additive constant on an open neighborhood of every s ∈ [0, 1], they must be equal up to additive constant throughout the interval, and requirement (1) above implies that this constant is zero.

A corollary of this discussion is that Ψ_{1}^{P} = Ψ_{1}^{P′} whenever P and P′ are homotopically equivalent paths in T . To obtain this, it is enough to observe that if P^{r} is a homotopy, with r ∈ [0, 1] and P = P^{0}, P′ = P^{1}, then for each r ∈ [0, 1] we have P^{r} ∼ P^{r′} for all r′ in some neighborhood of r. This implies that Ψ_{1}^{P^{r}} (as a function of r) is constant on a neighborhood of each point in [0, 1], hence constant throughout [0, 1].

Since we are assuming X ∖ Y is simply connected, we can fix a point x_{0} ∈ X ∖ Y and define the pair (a_{1}(x), a_{2}(x)) to be the value (a_{1}(1), a_{2}(1)) obtained above by taking P (s) = (g(p(s)), h(p(s))) where p : [0, 1] → X is any path from x_{0} to x. For y in some neighborhood of each x ∈ X ∖Y, and for some t ∈ T, we have that a_{1}(y) is an affine function of d_{T} (t, g(y)). For this, it suffices to take a neighborhood and a t such that t ∈ I(P (s)) throughout that neighborhood and use requirement (2) above. In this neighborhood, T - comparison implies that a_{1} satisfies comparison with distance functions from below and a_{2} satisfies comparison with distance functions from above.

Now let C be the Lipschitz constant of f, and for δ > 0 write

U_{δ} =^{def} {x : a_{1}(x) < a_{2}(x) − 2Cδ} .

We claim that the argument in the paragraph just above implies that a_{2} satisfies comparison with distance functions from above on B(x, δ) for any x ∈ U_{δ}. To see this, first observe that the fact that a_{2} is Lipschitz with constant at most C implies that B(x, δ) ⊆ X ∖ Y, and then take t to be the midpoint in T between g(x) and h(x), noting that since g and h are both Lipschitz with constant C, we have that t is on the geodesic between g(x′) and h(x′) for all x′ ∈ B(x, δ).

We may now apply Lemma 9 (where the Y of the lemma statement is chosen so that U_{δ} = X ∖ Y) to see that a_{2} ⩽ ã_{2} on U_{δ}, where ã_{2} is the AMLE of the restriction of a_{2} to ∂U_{δ}. By symmetry, we may apply the same arguments to −a_{1} to obtain that a_{1} ⩾ ã_{1} on U_{δ}. Since a_{1} < a_{2} by construction, we now have

ã_{1} ⩽ a_{1} < a_{2} ⩽ ã_{2}.  (19)

Now it is a standard fact (see [4]) about AMLE that the suprema and infima of a difference of AMLEs (in this case ã_{2} − ã_{1}) is obtained on the boundary set (in this case ∂U_{δ}). This implies that ã_{1} ⩽ ã_{2} ⩽ ã_{1} + 2Cδ throughout the set U_{δ}. By (19), we now have that

sup_{x∈U_{δ}} |a_{1}(x) − a_{2}(x)| ⩽ 2Cδ

This implies sup_{x∈X} |a_{1}(x)−a_{2}(x)| ⩽ 2Cδ, and since this holds for all δ > 0, we have a_{1} = a_{2} throughout X, a contradiction. □

4. Proofs of Theorem 2 and Theorem 3

We first present the simple proof of Theorem 2, i.e, the local-global result for tree-valued ∞-harmonic functions. The proof is a modification of an argument in [42] from the setting of real-valued mappings to the setting of tree-valued mappings.

Proof of Theorem 2. Let U ⊆ G ∖ Ω be an open subset of G. Denote

L = Lip\overline{U}(f) = max_{\begin{matrix}x,y∈\overline{U} \\ x≠y \end{matrix}} \frac{d_{T} (f(x), f(y))}{d_{G}(x, y)}.  (20)

Let x, y ∈ \overline{U} be points at which the maximum in (20) is attained and d_{G}(x, y) is maximal among all such points. We will be done if we show that x, y ∈ ∂U. Assume for the sake of contradiction that x ∈ U (the case y ∈ U being similar).

If x is in the interior of an edge of G, we could move x slightly along the edge and increase d_{G}(x, y) without decreasing d_{T} (f(x), f(y))/d_{G}(x, y). So, assume that x ∈ V. The fact that f is ∞-harmonic on V ∖Ω ⊇ {x} means that there exist u, v ∈ N_{G}(x) with d_{T} (f(u), f(v)) = 2L and f(x) is a midpoint between f(u) and f(v) in T. Since x ∈ U (and f is linear on the edges of G) there exists ε > 0 and z_{u}, z_{v} ∈ U such that z_{u} ∈ xu ∈ E, z_{v} ∈ xv ∈ E, d_{G}(x, z_{u}) = d_{G}(x, z_{v}) = ε and f(z_{u}) (resp. f(z_{v})) is the point on the geodesic in T joining f(x) and f(u) (resp. f(v)) at distance Lε from f(x). Because T is a metric tree, either d_{T} (f(y), f(z_{u})) = d_{T} (f(x), f(y)) + Lε or d_{T} (f(y), f(z_{v})) = d_{T} (f(x), f(y)) + Lε. Assume without loss of generality that d_{T} (f(y), f(z_{u})) = d_{T} (f(x), f(y))+Lε = L(d_{G}(x, y)+ε). Then d_{G}(z_{u}, y) = d_{G}(x, y) + ε and d_{T} (f(z_{u}), f(y))/d_{G}(z_{u}, y) = L, contradicting the maximality of d_{G}(x, y). □

As in the classical proof that metric convexity and the binary intersection property implies the isometric extension property (see [7, Prop. 1.4]), for the proof of Theorem 3 we shall construct f̃ by extending to one additional point at a time. The proof of Theorem 3 relies on a specific choice of the ordering of the points for the purpose of such a point-by-point construction. Our argument uses a variant of an algorithm from [28].

Proof of Theorem 3. Write |V ∖ Ω| = n. We shall construct inductively a special ordering w_{1}, …, w_{n} of the points of V ∖ Ω, and extend f to these points one by one according to this ordering. Assume that w_{1}, …, w_{k} have been defined, as well as the values f̃(w_{1}), …, f̃(w_{k}) ∈ Z (if k = 0 this assumption is vacuous).

Write Ω_{0} = Ω and Ω_{k} = Ω ∪ {w_{1}, …, w_{k}}. Given distinct x, y ∈ V we shall say that x_{0}, x_{1}, … x_{ℓ} ∈ V is a path joining x and y which is external to Ω_{k} if x_{0} = x, x_{ℓ} = y, and for all i ∈ {0, …, ℓ − 1} we have x_{i}x_{i+1} ∈ E and {x_{i}, x_{i+1}} ⊈ Ω_{k}. Let d_{k}(x, y) be the minimum over ℓ ∈ ℕ such that there exists a path x_{0}, x_{1}, … x_{ℓ} ∈ V joining x and y which is external to Ω_{k}. If no such path exists we set d_{k}(x, y) = ∞. We also set d_{k}(x, x) = 0 for all x ∈ V. Then d_{k} : V × V → {0} ∪ ℕ ∪ {∞} clearly satisfies the triangle inequality and d_{k}(·, ·) ⩾ d_{G}(·, ·) pointwise.

We distinguish between two cases: Case 1. For all distinct x, y ∈ Ω_{k} we have d_{k}(x, y) = ∞. In this case order the points of V ∖ Ω_{k} arbitrarily, i.e., V ∖ Ω_{k} = {w_{k+1}, …, w_{n}}. If w ∈ {w_{k+1}, …, w_{n}} then by the connectedness of G, there exists a path in G joining w and some point x_{w} ∈ Ω_{k}. Note that x_{w} is uniquely determined by w, since if there were another path joining w and some point y_{w} ∈ Ω_{k} which isn’t x_{w} then d_{k}(x_{w}, y_{w}) < ∞, contradicting our assumption in Case 1. We can therefore define in this case f̃(w) = f̃(x_{w}).

Case 2. For some distinct x, y ∈ Ω_{k} we have d_{k}(x, y) < ∞. In this case define

L_{k} = max_{\begin{matrix}x,y∈Ω_{k} \\ x≠y \end{matrix}} \frac{d_{Z} (f̃(x), f̃(y))}{d_{k}(x, y)}.  (21)

Our assumption implies that L_{k} > 0. Choose x, y ∈ Ω_{k} that are distinct and satisfy L_{k}d_{k}(x, y) = d_{Z} (f̃(x), f̃(y)). Write ℓ = d_{k}(x, y) and let x_{0}, x_{1}, … x_{ℓ} ∈ V be a path joining x and y which is external to Ω_{k}. Then x_{1} ∉ Ω_{k}, so we may define w_{k+1} = x_{1}. We claim that

⋂_{a∈Ω_{k}} B_{Z} (f̃(a), L_{k}d_{k}(a, w_{k+1})) ≠ ∅.  (22)

To prove (22), by the fact that Z has the binary intersection property, it suffices to show that for all a, b ∈ Ω_{k} we have

B_{Z} (f̃(a), L_{k}d_{k}(a, w_{k+1})) ∩ B_{Z} (f̃(b), L_{k}d_{k}(b, w_{k+1})) ≠ ∅.  (23)

If either d_{k} (a, w_{k+1}) = ∞ or d_{k} (b, w_{k+1}) = ∞ then (23) is trivial. Assume therefore that d_{k} (a, w_{k+1}) and d_{k} (b, w_{k+1}) are finite. Define λ ∈ [0, 1] by

λ = \frac{d_{k} (a, w_{k+1})}{d_{k} (a, w_{k+1}) + d_{k} (b, w_{k+1})}.  (24)

Since Z is metrically convex, there exists a point z ∈ Z such that

d_{Z} (z, f̃(a)) = λd_{Z} (f̃(a), f̃(b)) and d_{Z} (z, f̃(b)) = (1 − λ)d_{Z} (f̃(a), f̃(b)) . (25)

The definition of L_{k} implies

d_{Z} (f̃(a), f̃(b)) ⩽ L_{k}d_{k}(a, b) ⩽ L_{k}(d_{k}(a, w_{k+1}) + d_{k}(b, w_{k+1})).  (26)

Using (24) and (25), we deduce from (26) that

d_{Z} (z, f̃(a)) ⩽ L_{k}d_{k}(a, w_{k+1}) and d_{Z} (z, f̃(b)) ⩽ L_{k}d_{k}(b, w_{k+1}),

proving (23). Having proved (22), we let f̃(w_{k+1}) be an arbitrary point satisfying

f̃(w_{k+1}) ∈ ⋂_{a∈Ω_{k}} B_{Z} (f̃(a), L_{k}d_{k}(a, w_{k+1})) .  (27)

The above inductive construction produces a function f̃ : V → Z that extends f. We claim that f̃ is ∞-harmonic on V ∖ Ω. To see this note that for all x, y ∈ V the sequence {d_{k}(x, y)}_{k=1}^{n} ⊆ {0} ∪ ℕ ∪ {∞} is non-decreasing. We shall next show that the sequence {L_{k}}_{k=1}^{n}, defined in (21), is non-increasing. Indeed, assume that L_{k+1} > 0 and take distinct a, b ∈ Ω_{k+1} such that L_{k+1}d_{k+1}(a, b) = d_{Z} (f̃(a), f̃(b)). If a, b ∈ Ω_{k} then it follows from the definition of L_{k} that L_{k+1} ⩽ L_{k}, since d_{k+1}(a, b) ⩾ d_{k}(a, b). By symmetry, it remains to deal with the case a ∈ Ω_{k} and b = w_{k+1}. In this case, since by our construction we have f̃(w_{k+1}) ∈ B_{Z} (f̃(a), L_{k}d_{k}(a, w_{k+1})) ⊆ B_{Z} (f̃(a), L_{k}d_{k+1}(a, w_{k+1})), it follows once more that L_{k+1} ⩽ L_{k}.

Fix k ∈ {0, …, n − 1}. If f̃(w_{k+1}) was defined in Case 1 of our inductive construction, then f̃ is constant on N_{G}(w_{k+1}) ∪ {w_{k+1}}, in which fact the ∞-harmonic conditions (6), (7) for f̃ at w_{k+1} hold trivially. If, on the other hand, f̃(w_{k+1}) was defined in Case 2 of our inductive construction, then there exist distinct x, y ∈ Ω_{k} with L_{k}d_{k}(x, y) = d_{Z} (f̃(x), f̃(y)), such that for ℓ = d_{k}(x, y) there are x_{0}, x_{1}, … x_{ℓ} ∈ V which form a path joining x and y which is external to Ω_{k}, and x_{1} = w_{k+1}. For every i ∈ {1, …, ℓ − 1} either x_{i} ∉ Ω_{k} or x_{i+1} ∉ Ω_{k}, and therefore at least one of the values f̃(x_{i}), f̃(x_{i+1}) was defined after stage k + 1 of our inductive construction. This means that for some j ⩾ k we have |Ω_{j} ∩ {x_{i}, x_{i+1}}| = 1 and

d_{Z} (f̃(x_{i}), f̃(x_{i+1})) ⩽ L_{j}d_{j}(x_{i}, x_{i+1}) = L_{j} ⩽ L_{k},  (28)

where we used the fact that x_{i}x_{i+1} ∈ E, and therefore, since |Ω_{j} ∩ {x_{i}, x_{i+1}}| = 1, the path x_{i}, x_{i+1} is external to Ω_{j}. Thus

L_{k}ℓ = d_{Z} (f̃(x), f̃(y))
⩽ d_{Z} (f̃(x_{0}), f̃(x_{2})) + d_{Z} (f̃(x_{2}), f̃(x_{ℓ}))
⩽ d_{Z} (f̃(x_{0}), f̃(x_{2})) + ∑_{i=2}^{ℓ−1} d_{Z} (f̃(x_{i}), f̃(x_{i+1}))

⩽^{(28)} d_{Z} (f̃(x_{0}), f̃(x_{1})) + d_{Z} (f̃(x_{1}), f̃(x_{2})) + L_{k}(ℓ − 2)
⩽^{(28)} L_{k}ℓ.  (29)

It follows that all the inequalities in (29) actually hold as equality. Therefore we have d_{Z} (f̃(x), f̃(w_{k+1})) = d_{Z} (f̃(w_{k+1}), f̃(x_{2})) = L_{k} and d_{Z} (f̃(x), f̃(x_{2})) = 2L_{k}. Since by

construction x, x_{2} ∈ N_{G}(w_{k+1}), in order to show that f̃ is ∞-harmonic at w_{k+1} it remains to check that for all u ∈ N_{G}(w_{k+1}) we have d_{Z} (f̃(u), f̃(w_{k+1})) ⩽ L_{k}. But, our construction

ensures that for some j ⩾ k we have d_{Z} (f̃(u), f̃(w_{k+1})) ⩽ L_{j}d_{j}(u, w_{k+1}) = L_{j} (using uw_{k+1} ∈ E), and the required result follows since L_{j} ⩽ L_{k}.

Denote L = Lip_{Ω}(f). In order to complete the proof of Theorem 3 it remains to argue that Lip_{V} \binom{}{f} = L. We will do so by proving by induction on k that Lip_{Ω_{k}} \binom{}{f} = L.

For k = 0 this is the definition of L, so assuming that Lip_{Ω_{k}} \binom{}{f} = L we need to deduce

that if x ∈ Ω_{k} then d_{Z} (f̃(x), f̃(w_{k+1})) ⩽ Ld_{G}(x, w_{k+1}). Denote ℓ = d_{G}(x, w_{k+1}) and let w_{k+1} = y_{0}, y_{1}, …, y_{ℓ} = x ∈ V satisfy y_{i}y_{i+1} ∈ E for all i ∈ {0, …, ℓ − 1}. Let j ∈ {1, …, ℓ} be the smallest index such that y_{j} ∈ Ω_{k}. Since y_{0}, y_{1}, …, y_{ℓ} is a shortest path in G joining w_{k+1} and x, and the path w_{k+1} = y_{0}, …, y_{j} is external to Ω_{k} by the definition of j, it follows that d_{k}(w_{k+1}, y_{j}) = d_{G}(w_{k+1}, y_{j}) = j and d_{G}(y_{j}, x) = ℓ − j. Since y_{j}, x ∈ Ω_{k}, by the inductive hypothesis we have d_{Z} (f̃(y_{j}), f̃(x)) ⩽ L(ℓ − j). Also, it follows from (27)

that d_{Z} (f̃(w_{k+1}), f̃(y_{j})) ⩽ L_{k}d_{k}(y_{j}, w_{k+1}) = L_{k}j ⩽ Lj, where we used the fact that since

d_{k}(·, ·) ⩾ d_{G}(·, ·), it follows from (21) and the inductive hypothesis that L_{k} ⩽ Lip_{Ω_{k}} \binom{}{f} = L. So, in conclusion,

d_{Z} (f̃(x), f̃(w_{k+1})) ⩽ d_{Z} (f̃(w_{k+1}), f̃(y_{j})) + d_{Z} (f̃(y_{j}), f̃(x)) ⩽ Lj + L(ℓ − j) = L_{ℓ}. □

5. Existence

Here we prove the existence part of Theorem 1, i.e., we establish the following:

Theorem 13. Let (X, d_{X}) be a locally compact length space and (T, d_{T}) a metric tree. Then for every closed Y ⊆ X, every Lipschitz mapping f : Y → T admits an AMLE.

Proof. Assume first that X is compact. We will construct an AMLE f̃ of f as a limit of discrete approximations.

For each ε ∈ (0, 1/4), let Λ_{ε} be a finite subset of X such that

X ⊆ ⋃_{x∈Λ_{ε}} B_{X}(x, ε) and Y ⊆ ⋃_{y∈Λ_{ε}∩Y} B_{X}(y, ε).  (30)

Let G_{ε} be the graph whose vertices are the elements of Λ_{ε}, with x, y ∈ Λ_{ε} adjacent when d_{X}(x, y) ⩽ \sqrt{ε}.

For any x and y in Λ_{ε}, we can find an arbitrarily-close-to-minimal length path between them and a sequence of points x = x_{0}, x_{1}, x_{2}, …, x_{k} = y spaced at intervals of \sqrt{ε} − 2ε along the path, where k − 1 is the integer part of d_{X}(x, y)/(\sqrt{ε} − 2ε), and can then find points x̃_{i} ∈ B(x_{i}, ε) ∩ Λ_{ε}. Since d(x_{i}, x_{i+1}) ⩽ \sqrt{ε} we conclude that d_{G_{ε}}(x, y) ⩽ k. It is also clear that d_{G_{ε}}(x, y) ⩾ d(x, y)/\sqrt{ε}. Hence,

|d_{G_{ε}}(x, y)\sqrt{ε} − d_{X}(x, y)| ⩽ C\sqrt{ε},  (31)

where C depends only on the diameter of X.

Let f̃_{ε} be an ∞-harmonic extension of f|_{Y ∩Λ_{ε}} to all of G_{ε}, the existence of which is due to Theorem 3 (since T is a 1-absolute Lipschitz retract). Note that on Λ_{ε} we have the pointwise inequality d_{X}(·, ·) ⩽ \sqrt{ε}d_{G_{ε}}(·, ·). It follows that the Lipschitz constant of f|_{Y ∩Λ_{ε}} with respect to the metric \sqrt{ε}d_{G_{ε}} is bounded above by Lip_{Y} (f), and hence the Lipschitz constant of f̃_{ε} with respect to the metric \sqrt{ε}d_{G_{ε}} is also bounded above by Lip_{Y} (f).

Let N_{ε} ⊆ Λ_{ε} be a \sqrt{ε}-net in (Λ_{ε}, d_{X}), i.e., a maximal subset of Λ_{ε}, any two elements of which are separated in the metric d_{X} by at least \sqrt{ε}. For any distinct x, y ∈ N_{ε} we have

d_{T} (f̃_{ε}(x), f̃_{ε}(y)) ⩽ Lip_{Y} (f)\sqrt{ε}d_{G_{ε}}(x, y)

⩽^{(31)} Lip_{Y} (f) (d_{X}(x, y) + C\sqrt{ε}) ⩽ Lip_{Y} (f)(1 + C)d_{X}(x, y). (32)

It follows that we can extends f̃|_{N_{ε}} to a function f_{ε}^{∗} : X → T that is Lipschitz with constant

Lip_{Y} (f)(1 + C) (this extension can be done in an arbitrary way, using the fact that T is a 1-absolute Lipschitz retract). Since the functions f_{ε}^{∗} are equicontinuous, the Arzela-Ascoli Theorem [38, Thm. 6.1] says that there exists a subsequence {ε_{n}}_{n=1}^{∞} ⊆ (0, 1/4) tending to zero such that f_{ε_{n}}^{∗} converges uniformly to f^{∗} : X → T. We aim to show that f^{∗} is an AMLE of f.

By Proposition 7 it is enough to show that for each t ∈ T and open W ⊆ X∖Y, z ∈ X∖W, b ⩾ 0 and c ∈ ℝ, we have the following:

∀x ∈ ∂W d_{T} (t, f^{∗}(x)) ⩽ bd_{X}(x, z) + c ⟹ ∀x ∈ W d_{T} (t, f^{∗}(x)) ⩽ bd_{X}(x, z) + c. (33)

By uniform convergence, for every δ > 0 there exists n_{0} ∈ ℕ such that for every n ⩾ n_{0} if for every x ∈ ∂W we have

d_{T} (t, f^{∗}(x)) ⩽ bd_{X}(x, z) + c,  (34)

then for every x ∈ ∂W we have

d_{T} (t, f_{ε_{n}}^{∗}(x)) ⩽ bd_{X}(x, z) + c + δ.  (35)

Assume from now on that (34) holds for all x ∈ ∂W. Let V_{ε_{n}} ⊆ N_{ε_{n}} be the set of u ∈ N_{ε_{n}} ⊆ Λ_{ε_{n}} for which there exists w ∈ W such that d_{X}(u, w) ⩽ \sqrt{ε_{n}}. Define W_{ε_{n}} to be the open subset of the 1-dimensional simplicial complex corresponding to the graph G_{ε_{n}} consisting of the union of all the half-open intervals [u, v), where u, v ∈ Λ_{ε_{n}}, uv is an edge of G_{ε_{n}} and u ∈ V_{ε_{n}}. Any point v ∈ ∂W_{ε_{n}} of the boundary of W_{ε_{n}} in G_{ε_{n}} is at d_{X}-distance greater than \sqrt{ε_{n}} from W, but at d_{X}-distance at most \sqrt{ε_{n}} from some point of N_{ε_{n}} whose d_{X}-distance from W is at most \sqrt{ε_{n}}. Thus

v ∈ ∂W_{ε_{n}} ⟹ d_{X}(u, ∂W) ⩽ 2\sqrt{ε_{n}}.  (36)

Let z_{ε_{n}} be any one of the d_{X}-closest points of z in N_{ε_{n}}. By (30) and the definition of N_{ε_{n}} we have

d_{X}(z_{ε_{n}}, z) ⩽ \sqrt{ε_{n}} + ε_{n} ⩽ 2\sqrt{ε_{n}}.  (37)

Since f_{ε_{n}}^{∗} is Lipschitz with constant Lip_{Y} (f)(1 + C), for every v ∈ ∂W_{ε_{n}} we have

d_{T} (t, f_{ε_{n}}^{∗}(v)) ⩽^{(35)∧(36)} bd_{X}(v, z) + c + δ + 2Lip_{Y} (f)(1 + C)\sqrt{ε_{n}}
⩽^{(37)} bd_{X} (v, z_{ε_{n}}) + c + δ + 2(Lip_{Y} (f)(1 + C) + b)\sqrt{ε_{n}}
⩽^{(31)} b\sqrt{ε_{n}}d_{G_{εn}} (v, z_{ε_{n}}) + c + δ + (2Lip_{Y} (f)(1 + C) + 2b + bC)\sqrt{ε_{n}}
⩽ b\sqrt{ε_{n}}d_{G_{εn}} (v, z_{ε_{n}}) + c + δ + K\sqrt{ε_{n}},  (38)

where K > 0 is independent of n. Observe that if z_{ε_{n}} ∈ W_{ε_{n}} then since z ∉ W we have d_{X}(z_{ε_{n}}, ∂W) ⩽ 6\sqrt{ε_{n}}. In this case the same argument as above shows that (38) holds for v = z_{ε_{n}} as well (with a different value of K). Thus, the bound (38) holds for all v ∈ ∂(W_{ε_{n}} ∖ {z_{ε_{n}}}). By Theorem 2 and Proposition 7, it follows that for every v ∈ W_{ε_{n}} ∖ {z_{ε_{n}}}, and hence also for all v ∈ V_{ε_{n}}, we have

d_{T} (t, f_{ε_{n}}^{∗}(v)) ⩽ b\sqrt{ε_{n}}d_{G_{εn}} (v, z_{ε_{n}}) + c + δ + K\sqrt{ε_{n}}

⩽^{(31)∧(37)} bd_{X} (v, z) + c + δ + (K + Cb + 2b)\sqrt{ε_{n}}. (39)

Since any point of W is at d_{X}-distance at most ε_{n} + \sqrt{ε_{n}} ⩽ 2\sqrt{ε_{n}} from V_{ε_{n}}, and since f_{ε_{n}}^{∗} is Lipschitz with constant independent of n, we see from (39) that for some K′ > 0 independent of n, for all x ∈ W we have:

d_{T} (t, f_{ε_{n}}^{∗}(x)) ⩽ bd_{X} (x, z) + c + δ + K′\sqrt{ε_{n}}.  (40)

Letting n tend to ∞ in (40), it follows that

d_{T} (t, f^{∗}(x)) ⩽ bd_{X} (x, z) + c + δ.  (41)

Since (41) holds for all δ > 0, we have proved the desired implication (33).

When X is locally compact but not necessarily compact, the proof of Theorem 13 follows from a direct reduction to the compact case. Indeed, by Remark 2 it suffices to prove (33) when b > 0. In this case, since T is bounded, the upper bounds in (33) are trivial if d_{X}(x, z) is sufficiently large. Thus, it suffices to prove (33) for the intersection of W with a large enough ball centered at z. □

6. Politics

In this section we prove Proposition 4. We require some notation (in particular, a definition of value) to make the statement of Proposition 4 precise.

A strategy for a player is a way of choosing the player’s next move as a function of all previously played moves and all previous coin tosses. It is a map from the set of partially played games to moves (or in the case of a random strategy, a probability distribution on moves). We might expect a good strategy to be Markovian, i.e., a map from the current state to the next move, but it is useful to allow more general strategies that take into account the history.

Given two strategies S_{I}, S_{II}, let F(S_{I}, S_{II}) be the expected total payoff (including the running payoffs received) when the players adopt these strategies. We define F to be some fixed constant C if the game does not terminate with probability one, or if this expectation does not exist.

The value of the game for player I is defined as sup_{S_{I}} inf_{S_{II}} F(S_{I}, S_{II}). The value for player II is inf_{S_{II}} sup_{S_{I}} F(S_{I}, S_{II}). The game has a value when these two quantities are equal. It turns out that Politics always has a value for any choice of initial states x_{0} ∈ V and t_{0} ∈ T; this is a consequence of a general theorem (since the payoff function is a zero-sum Borel-measurable function of the infinite sequence of moves [33]; see also [41] for more on stochastic games).

Proof of Proposition 4. First we introduce some notation: when the game position is at x_{k}, we let y_{k} and z_{k} denote two of the vertices adjacent to x_{k} that maximize d_{T} (f̃(x_{k}), f̃(·)),

chosen so that f̃(x_{k}) is the midpoint of f̃(y_{k}) and f̃(z_{k}). Write for x ∈ V,

δ(x) =^{def} sup_{y∈N_{G}(x)} d_{T} (f̃(x), f̃(y)) ,

and

M_{k} =^{def} δ(x_{k}) = d_{T} (f̃(x_{k}), f̃(y_{k})) = d_{T} (f̃(x_{k}), f̃(z_{k})) .

Using this notation, we now give a strategy for player II that makes d (f̃(x_{k}), t_{k}) plus the total payoff thus far for Player I a supermartingale. Player II always chooses t_{k} to be the element in {f̃(y_{k−1}), f̃(z_{k−1})} on which d_{T} (·, o_{k}) is largest; if she wins the coin toss, she then

chooses x_{k} to be so that f̃(x_{k}) is that element. To establish the supermartingale property, we must show that, regardless of player I’s strategy, we have

𝔼[d_{T} (o_{k}, t_{k}) − d_{T} (o_{k}, t_{k−1})] ⩾ 𝔼 [d_{T} (f̃(x_{k}), t_{k}) − d_{T} (f̃(x_{k−1}), t_{k−1})] .  (42)

It is not hard to see that we have deterministically

d_{T} (o_{k}, t_{k}) − d_{T} (o_{k}, t_{k−1}) ⩾ d_{T} (f̃(x_{k−1}), t_{k}) − d_{T} (f̃(x_{k−1}), t_{k−1}) .  (43)

Indeed, if o_{k} and t_{k−1} are in distinct components of T ∖ {f̃(x_{k−1})}, then the same will be true of o_{k} and t_{k}, and (43) holds as equality; if o_{k} and t_{k−1} are in the same component of T ∖ {f̃(x_{k−1})} then o_{k} and t_{k} will be in opposite components of T ∖ {f̃(x_{k−1})}, and the left hand side minus the right hand side of (43) becomes twice the distance from f̃(x_{k−1}) of the least common ancestor of o_{k} and and t_{k−1} in the tree rooted at f̃(x_{k−1}).

Due to (43), in order to prove (42) it is enough to show that

𝔼 [d_{T} (f̃(x_{k}), t_{k}) − d_{T} (f̃(x_{k−1}), t_{k})] ⩽ 0,

which is clear since if player II wins the coin toss this quantity will be −M_{k} and if player I wins the coin toss it will be at most M_{k}.

Next we give a very similar strategy for player I that makes d_{T} (f̃(x_{k}), t_{k}) plus the total payoff thus far for Player I a submartingale. In this strategy, Player I always chooses o_{k} to be the element in {f̃(y_{k−1}), f̃(z_{k−1})} on which d_{T} (·, t_{k−1}) is largest; if he wins the coin

toss, he then chooses x_{k} to be so that f̃(x_{k}) is that element. To establish the submartingale property, we must now show that

𝔼[d_{T} (o_{k}, t_{k}) − d_{T} (o_{k}, t_{k−1})] ⩽ 𝔼 [d_{T} (f̃(x_{k}), t_{k}) − d_{T} (f̃(x_{k−1}), t_{k−1})] .  (44)

Note that by strategy definition o_{k} is on the opposite side of f̃(x_{k−1}) from t_{k−1}, so we may write d_{T} (o_{k}, t_{k−1}) = M_{k} + d_{T} (f̃(x_{k−1}), t_{k−1}). Plugging this into (44), what we seek to show becomes

𝔼[d_{T} (o_{k}, t_{k}) − M_{k}] ⩽ 𝔼 [d_{T} (f̃(x_{k}), t_{k})] ,  (45)

which we see by noting that the right hand side of (45) is equal to d_{T} (o_{k}, t_{k}) when player I wins the coin toss (and makes f̃(x_{k}) = o_{k}) and at least d_{T} (o_{k}, t_{k}) − 2M_{k} when player II wins the coin toss, since d_{T} (f̃(x_{k}), o_{k}) ⩽ 2M_{k} for any valid choice of x_{k}.

To conclude the proof, we need to modify the strategy in such a way that forces the game to terminate without sacrificing the payoff expectation. If both players adopt the above strategy, it is clear that the increments d_{T} (f̃(x_{k−1}), f̃(x_{k})) are non-decreasing, and that the distance from any fixed endpoint of the tree has at least probability 1/2 of increasing at each step; from this, it follows that the length of game play is a random variable with exponential decay. If the other player makes other moves, which are not optimal from the point of view of optimizing the payoff, then we can wait until the cumulative amount the other player has “given up” is greater than twice the diameter of T, and then force the game to end by placing a target at a single point and subsequently always moving x_{k} closer to that point when winning a coin toss. (The loss from the sub-optimality of this strategy is less than the gain from the amount the other player gave up.) If a player adopts this strategy, then the total time duration of the game is a random variable whose law decays exponentially; this yields the uniform integrability necessary for the sub-martingale optional stopping theorem, which implies Proposition 4. □

Acknowledgements. A. N. is supported by NSF grants CCF-0635078 and CCF-0832795, BSF grant 2006009, and the Packard Foundation. S. S. is supported by NSF grants DMS-0645585 and OISE-0730136. We are grateful to the anonymous referee for many helpful comments and suggestions.

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Courant Institute, New York University

  E-mail address: naor@cims.nyu.edu

Massachusetts Institute of Technology

  E-mail address: sheffield@math.mit.edu