Chordal Hausdorff Convergence and Quasihyperbolic Distance

We study Hausdor convergence (and related topics) in the chordalization of a metric space to better understand pointed Gromov-Hausdor convergence of quasihyperbolic distances (and other conformal distances).


Introduction
Here we examine assorted types of convergence of sets with an eye towards securing the convergence of an associated sequence of metric spaces de ned via conformal metrics. The starting point for our work is the following folklore fact, which holds because the Euclidean distances dist(x, ∂Ω i ) converge uniformly in R n to dist(x, ∂Ω).
Let Ω, Ω i R n be domains. Suppose the closed sets R n \ Ω i coverge, with respect to Hausdor distance in R n , to R n \ Ω. Then the quasihyperbolic distances in Ω i converge locally uniformly in Ω to quasihyperbolic distance in Ω. We present various and sundry generalizations of this result. In particular, we believe that the correct notion of convergence (for the associated distances) is that of pointed Gromov-Hausdor convergence, a concept that we review in §2. 3. To a certain extent, this work is about the pointed Gromov-Hausdor convergence of quasihyperbolic metric spaces. In this setting compactness can be weakened to just boundedness, and we introduce the notion of bounded uniform convergence which replaces local uniform convergence; see §3.1. This is especially useful when local compactness is not available, e.g., in the Banach space setting.
A non-complete locally complete recti ably connected metric space Ω is dubbed a quasihyperbolic space. Each such Ω carries a quasihyperbolic metric δ − ds = δ − ∂Ω ds, whose length distance k = k Ω is called quasihyperbolic distance in Ω; here δ(x) = δ ∂Ω (x) := dist(x, ∂Ω) is the distance from x to the metric boundary ∂Ω of Ω. See §2.4.1 for more details. Using terminology coined by Bonk-Heinonen-Koskela, the quasihyperbolization of Ω is the metric space Ω k := (Ω, k). A simple, but nonetheless important, special case of a quasihyperbolic space is any open connected proper subset of Euclidean space R n (with its induced Euclidean distance), and we call such an Ω R n an Euclidean quasihyperbolic space.
Since its introduction in the 1970's by Gehring and Palka, see [9], the quasihyperbolic metric has proven to be an especially useful and important tool in many areas of geometric analysis including classical geometric function theory, quasiconformal mapping theory, potential theory, complex dynamics, and even geometric group theory. Its importance, especially with regards to the program of 'metric space analysis', cannot be overstated.
Thus it is worthwhile to know when we can approximate the quasihyperbolization of a space by simpler spaces. That is, given a quasihyperbolic space Ω, when can we exhibit a sequence (Ω i ) of 'simple' quasihyperbolic spaces whose quasihyperbolizations (Ω i , k i ) converge, somehow, to (Ω, k)? For example, a quasihyperbolic space with a nite boundary is 'simple'. Such a program is at the heart of recent work by Väisälä and Luiro (see [26] and [19]) who approximate quasihyperbolic plane domains by punctured planes with nite boundaries.
Here is a special case of our Theorem 4.11, which in turn relies heavily on Theorem 4.4.
Theorem. Suppose a sequence (A i ) of closed subsets ofR n converges, with respect to chordal Hausdor distance, to a closed set A ≠ {∞}. Then, with respect to pointed Gromov-Hausdor distance, the quasihyperbolizations of (R n \ A i ) converge to the quasihyperbolization of R n \ A.
We also establish analogs of the above for other conformal metrics including the Ferrand and Kulkarni-Pinkhall-Thurston metrics (on quasihyperbolic domains inR n ) as well as the hyperbolic metric (on hyperbolic domains inĈ). See §4.2 and §4. 3. In Theorem 4.11, we generalize the above replacing R n by an arbitrary complete length space. Here Theorem 3.1 plays a crucial role; roughly speaking, when closed sets chordal Hausdor converge, their associated distance functions converge uniformly on bounded sets.
In the pointed Gromov Hausdor setting, our notion of uniform convergence on bounded sets is the natural analog of uniform convergence on compacts sets. Just as in the Euclidean setting where the chordal distance induces the one-point compacti cation, the chordalization of a metric space (see §2.2) produces a bounded distance whose topology agrees with the one-point extension topology. In general, there need be no compactness nor even local compactness, and we expect these ideas to prove useful in other situations.
In Theorems 3.1, 3.5 and Corollary 3.6 we show that chordal Hausdor convergence implies bounded uniform convergence of distances, investigate the consequences of such, and characterize chordal Hausdor convergence in terms of bounded uniform convergence. One consequence of Corollary 3.6 is the nite approx-Both inequalities above can be strict, but if X is a complete length space, then δ ∂Ω = δ X\Ω . A metric space X is locally complete provided each point is an interior point of some complete subspace; that is, for each x ∈ X there is a complete C ⊂ X with x ∈ int(C). Since closed subspaces of complete spaces are complete, it is not hard to check that this is the same as requiring that for all x ∈ X, dist(x, ∂X) > . (Equivalently, ∂X is closed inX, or each point has an open neighborhood whose closure is complete.) For example, this holds when X is locally compact.
When d is some other distance on X,X d and ∂ d X :=X d \ X denote the metric completion and metric boundary, respectively, of X d := (X, d). Also, B d (x; r) and S d (x; r) are the open ball and sphere (of radius r centered at the point x) in X d , and int d (A), cl d (A), bd d (A) are the interior, closure, boundary (respectively) of A in X d . For example, when X is recti ably connected, d could be the intrinsic length distance l = l X on X. A path in X is a continuous map R ⊃ I γ − → X where I = Iγ is an interval (called the parameter interval for γ) that may be closed or open or neither and nite or in nite. The trajectory of such a path γ is |γ| := γ(I) which we call a curve. When I is closed and I ≠ R, ∂γ := γ(∂I) denotes the set of endpoints of γ which consists of one or two points depending on whether or not I is compact.
We call γ a compact path if its parameter interval I is compact (which we often assume to be [ , ]).
When ∂γ = {a, b}, we write γ : a b (in X) to indicate that γ is a path (in X) with initial point a and terminal point b; this notation is also meant to imply an orientation-a precedes b on γ.
We note that every compact path contains an arc with the same endpoints; see [23].
Euclidean n-dimensional space is denoted R n and its one point extension isR n := R n ∪ {∞} together with the Euclidean chordal distance Always χ(x, y) ≤ |x − y|, so the "identity" inclusion R n id →R n is -Lip. Also, Thus R n id →R n is locally bi-Lipschitz, and so an embedding.
We refer to [2] for the de nition and basic properties of Möbius transformations. We introduce the following notation: We remind the reader that the inversion J, ofR n with respect to the origin, can be viewed as re ection across the unit sphere. However, the inversion Jp is the translation x → x − p followed by re ection across the unit sphere. Thus, e.g., J − = J whereas J − p (y) = y * + p. We utilize the following information about Möbius transformations; see [ For later use, we record the following.

Lemma. The Lipschitz constant H(x) := H(Jx) for Jx is given by
The Poincaré extensions of the maps J and Tx are J forR n+ and translation by −x := (−x, ) ∈ R n × R. Thus the Poincaré extension of Jx is J x , and writing x := e n+ − x = (−x, ) we obtaiñ and so

. Metric Chordal Distance
The one-point extensionX of X is de ned viâ which is de ned for all x, y ∈ X. Sometimes this is a distance function, but in general it may not satisfy the triangle inequality, so we de nê Then for all x, y ∈ X, In particular,d is a distance function on X and the map (X, |·|) id → (X,d) is: 1-Lipschitz, locally bilipschitz, and a t-quasimöbius homeomorphism. Moreover, when our original space X is unbounded, there is a unique point in the completion of (X,d) that corresponds to the point ∞ inX and the distance functiond on X extends in the usual way toX. Also, the metric topology induced byd onX is precisely the one-point extension topology.
We calld the chordal distance onX, and (X,d) is the chordalization of (X, |·|).² Thed-metric quantities inX are denoted either by attachingd or by using aĥat. For example: Bd(x; r) is ad-ball centered at x, and dist(x, A),diam(A),ˆ (γ) are thed-distance from a point to a set, thed-diameter of a set and thed-length of a path, respectively. The inequalities in (2.3) continue to hold for all points inX.
As (X, |·|) id → (X,d) is an embedding, whenever (x i ) is a sequence inX and z ∈ X we have lim i→∞d (x i , z) = ⇐⇒ x i ∈ X for all su ciently large i and lim Here is a useful elementary fact concerningX and the chordal distanced.

Lemma.
Let R > and put r :

. Hausdor and Gromov-Hausdor Distances
Here we recall the de nitions of Hausdor distance, Gromov-Hausdor distance, pointed Gromov-Hausdor distance, and state other relevant information.
2.6 Fact. For any non-empty A, B ⊂ X, the following quantities are all equal: The Hausdor distance dist H (A, B) = dist X H (A, B) between two non-empty subsets A, B of X is de ned to be the common value of the quantities listed in Fact 2.6.
A distance function δ on the disjoint union X Y of two metric spaces is admissible if its restriction to each of X, Y agrees with the original distances on X, Y respectively. Given t > , a distance function δ is t- where dist δ H denotes the Hausdor distance de ned on non-empty subsets of (X Y , δ). The Gromov-Hausdor distance between two non-empty metric spaces X and Y is A pointed metric space is a triple (X, d; a), that we often abbreviate as (X; a) when the distance function is understood, where (X, d) is a metric space and a is a xed base-point in X. Maps between pointed spaces are assumed to preserve base-points; thus a map f : (X; a) → (Y; b) satis es f (a) = b. Given t > and points We note that the above conditions imply that for all r ∈ ( , t − ], Following Gromov, we de ne the pointed Gromov-Hausdor distance between two pointed metric spaces (X; a) and (Y; b) via It is not di cult to see that Hausdor distance, Gromov-Hausdor distance, and pointed Gromov-Hausdor distance are all non-negative, symmetric, and satisfy the triangle inequality. While these are not true distance functions on all sets, it is well-known that: • dist H is a distance function on the class H(X) := C(X) ∩ B(X) of all non-empty closed bounded subsets of X, • dist GH is a distance function on the collection of all isometry classes of compact metric spaces, • dist GH * is a distance function on the collection of all isometry classes of pointed proper metric spaces.
See [11] and the many references mentioned therein.
Both Gromov-Hausdor distance and pointed Gromov-Hausdor distance are quantitatively equivalent to the existence of certain so-called rough isometries between the two spaces in question. Our terminology here is adopted from [6]. A map f : X → Y between metric spaces is an ε-rough isometric embedding provided |f (a) − b| < ε and B ⊂ N(fA; ε) .
We note that f need not be continuous, f (a) need not be b, and f (A) need not be a subset nor a superset of B.
For future reference we record the following information.
is a sequence of non-empty sets in X, and δ where dist H anddist H denote Hausdor distance in R n and inR n respectively.
(e) For any metric space X, both maps H(X)

. Conformal Metrics
A continuous function X ρ − → ( , ∞) on a recti ably connected metric space X induces a length distance dρ on X de ned by and where the in mum is taken over all recti able paths γ : a b in X. We describe this by calling ρ ds = ρ(x)|dx| a conformal metric on X; then ρ ds is complete if dρ is complete.
We call γ a ρ-geodesic if dρ(a, b) = ρ(γ); these need not be unique. We often write [a, b]ρ to indicate a ρ-geodesic with endpoints a, b, but one must be careful with this notation since these geodesics need not be unique. When we have a point x on a given xed geodesic [a, b]ρ, we write [a, x]ρ to mean the subarc of the given geodesic from a to x.
A simple but nonetheless important example is given by ρ = ; here we get the intrinsic length distance sometimes called the inner length distance. l = l X := d associated to the given distance on X, and then X l := (X, l). The l-geodesics are shortest paths in X, and X is a length space if (X, |·|) = (X, l).
We are primarily interested in the quasihyperbolic metric, discussed below, but also consider other conformal metrics on domains Ω ⊂R n . In this setting, if Ω contains the point at in nity, we must use local coordinates and remember that we are dealing with a metric; alternatively, we can work with the chordal (or the spherical) metric on Ω.
The metric ratio ρ ds/σ ds of two conformal metrics ρ ds and σ ds is a well-de ned positive function. We write ρ ≤ C σ to indicate that this metric ratio is bounded above by C.
We wish to extend this de nition to domains Ω in an ambient metric space X. With this in mind, we declare X to be a quasihyperbolic superspace provided X is complete, connected, and locally recti ably connected. For example, each complete length space is a quasihyperbolic superspace, as is each complete rectiably connected space X with the identity map X l → X a homeomorphism.

. . Estimates for QuasiHyperbolic Distance
Let X be a quasihyperbolic superspace, A X a non-empty closed subspace, and Ω a connected component Either inequality above may be strict; if X is a length space, equality holds for all x ∈ Ω.
Let Ω δ − → [ , +∞) denote any one of δ ∂Ω , δ A , δ X\Ω and consider the conformal metric δ − ds on Ω with its associated length distance d := d δ − .³ It is straightforward to verify that and thus for all a, b ∈ Ω,

. . Ferrand & Kulkarni-Pinkhall Metrics
These metrics, denoted φ ds and µ ds respectively, are de ned on each quasihyperbolic domain Ω inR n ; thus, Ω is an open connected subspace ofR n with Ω c :=R n \ Ω containing at least two points. The Ferrand metric φ ds = φ Ω ds on Ω can be de ned, for points x ∈ Ω ∩ R n , by The Kulkarni-Pinkall-Thurston metric µ ds = µ Ω ds on Ω can be de ned, for x ∈ Ω ∩ R n , by where λ B denotes the metric density for the hyperbolic metric λ B ds in the ball B. This metric was introduced by Ravi Kulkarni and Ulrich Pinkall in [17].
There is a method for calculating these metrics that is based on Euclidean diameters and circumdiameters. Recall that for any non-empty bounded set A ⊂ R n , there is a unique smallest closed (Euclidean) ball B 3 If δ = δ ∂Ω , d is quasihyperbolic distance in Ω, but this may not hold for the other two metrics.

. . The Poincaré Hyperbolic
Every hyperbolic plane domain carries a unique metric λ ds = λ Ω ds which enjoys the property that its pullback p * [λ ds], with respect to any holomorphic universal covering projection p : D → Ω, is the hyperbolic metric λ D (ζ )|dζ | = ( − |ζ | ) − |dζ | on the unit disk D. In terms of such a covering p, the (Euclidean) metricdensity λ = λ Ω of the Poincaré hyperbolic metric λ Ω ds can be determined from Yet another description is that λ ds is the unique maximal (or unique complete) metric on Ω that has constant Gaussian curvature − . In many cases (but certainly not all), the hyperbolic metric is bi-Lipschitz equivalent to the quasihyperbolic metric.

. . Metric Conventions
At times it is convenient to work with generalized distance functions that are allowed to take the value +∞. For example, if ∅ ≠ A R n is a closed set, then we can consider quasihyperbolic distance (or Ferrand or Kulkarni-Pinkall-Thurston distance) in R n \ A; here the distance between points from di erent components of R n \ A is +∞. This causes no di culties in the de nitions of Hausdor distance, Gromov-Hausdor distance, or pointed Gromov-Hausdor distance, although these quantities also could be in nite. Henceforth we allow such distance functions.
To minimize endless repetition, we also tacitly assume that whenever we mention some conformal distance, all necessary conditions on the underlying space hold. As an explicit example, the quasihyperbolic distance k = k U on U := X \ A is given by here X is assumed to be a quasihyperbolic superspace and ∅ ≠ A X a closed subspace.
Furthermore, we employ the following abbreviated terminology for pointed Gromov-Hausdor convergence. Given appropriate open subspaces U, U i of some metric space X, we write dist GH * (U i , k i ), (U, k) → provided each a ∈ U lies in U i for all but nitely many i, and when this is de ned, which will be true for all su ciently large i).
We adopt similar conventions for other conformal metrics.

. Chordal Hausdor Convergence
Everywhere, in this section, E bounded means that E is a bounded subset of X. Also, in general, A, A i are non-empty closed sets in X with chordal closuresÂ,Â i inX, and then δ, δ i andδ,δ i denote the distances to A, A i in X and toÂ,Â i inX (respectively). The notion of local uniform convergence, aka uniform convergence on compact subsets, is too restrictive for our purposes. We say that a sequence ( In particular, this implies that (a i ) is a bounded sequence in X.
For such an i large enough we would also havê .
It would follow that |a i − b| < ε/ , which would yield the contradiction Now we demonstrate that (δ i ) converges uniformly, in each bounded B ⊂ X, to δ. Let ε > be given. .
An easy compactness argument, in conjunction with Corollary 3.2, reveals that whendist H (Â i ,Â) → , each compact set in X \ A lies in all but nitely many X \ A i . However, no such result holds if we replace 'compact' with 'closed and bounded'.
Our next goal is to understand the consequences of (δ A i ) converging boundedly uniformly to δ A . The following preliminary result is useful.

Lemma. Let
Regardless of whether A is bounded or unbounded, Proof. Let δ := δ A and δ i := δ A i . To establish (3.3a), assume A is unbounded and let R > be given. Pick , sod(a i , ∞) < ε/ and a ∈ Bd(a i ; ε). Thus, with i := i ∨ i we see that (3.3b) holds whether A is bounded or unbounded. To establish (3.3c), let ε > be given. Put R := /ε. Choose i so that for all

Then δ
Below we describe exactly what it means for (δ A i ) to converge boundedly uniformly to δ A (when X is unbounded). As motivation for the following technical result, we mention (see Corollary 3.6 below) that . We now demonstrate that when (δ A i ) converges boundedly uniformly in X to δ A : Here is a more precise statement. See Example 3.22 for the importance of (δ A i ) being boundedly uniformly convergent. Also, recall Lemma 3.3.

Theorem. Let
Let ε > be given. According to Lemma 3.3(b), there exists an i so that for all i ≥ i , A ⊂ Nd(A i ; ε). We must establish a similar containment valid for A i for all large i.
Next we explain why A and A ∪ {∞} are the only possible subsequential limits for (Â i ) (as a sequence in (H(X),dist H )). To this end, suppose B ∈ H(X) (i.e., B is a non-empty closed subset ofX) and there exists a Here is a summary of parts of the above. Again, recall Lemma 3.3.

Corollary. For
In proper spaces we only require pointwise convergence (plus UBP) to obtain chordal Hausdor convergence.

Proposition.
Assume X is a proper metric space. The following are equivalent: Proof. That (a) =⇒ (b) follows from Corollary 3.6, and clearly (b) =⇒ (c) (with ρ = δ). Assume (c) holds. Since distance to a set is a 1-Lipschitz function, each δ i is 1-Lipschitz, so (δ i ) is equicontinuous and therefore we can apply Arzela-Ascoli (see for example [21]) to (δ i ) and each of its subsequences. In particular, by uniqueness of limits, (δ i ) converges locally uniformly-and hence as X is proper, boundedly uniformly-in X to ρ. To demonstrate thatdist H (Â i ,Â) → , we show that every subsequence of (Â i ) has a further subsequence that converges toÂ with respect todist H . Let (Â i j ) be such a subsequence. Since K(X) is compact, there is a subsubsequence (Â

. Carathéodory Convergence
Now we shift our focus to open sets (and initially X need only be a topological space). Let (U i ) be a sequence of open sets U i ⊂ X. Following [3], we de ne the core of (U i ) via this is always an open set but may be empty. We say that (U i ) Carathéodory converges to its core if and only if every subsequence (U i j ) of (U i ) has the same core. Beardon and Minda introduced the notion of Carathéodory core convergence and examined its relation to Carathéordory kernel convergence; see [3].
Since core(U i j ) ⊃ core(U i ) always holds, it is trivial that core(U i ) = X implies (U i ) converges to its core. Therefore if in nitely many U i equal X, then (U i ) converges to its core if and only if core(U i ) = X. Thus it is natural to assume that for all i, U i ≠ X, and that core(U i ) ≠ X. Now suppose that |·| is a compatible distance function on X, so the metric topology agrees with the original topology. Here is some information relating core(U i ) and the convergence of (U i ) to the behavior of the sequence of distances to X \ U i . Proof. (a) Suppose lim inf δ i = δ pointwise in X. Let x ∈ core(U i ). Then there are i ≥ and r > so that As this holds for all r ∈ ( , L), Evidently, equality holds at points in A. Assume X is proper, let x ∈ X \ A = U, and let r ∈ ( , δ(x)). Since B[x; r] is compact, it lies in all but nitely many U i , so B[x; r] ∩ A i = ∅ for all but nitely many i, whence lim inf δ i (x) ≥ r. Finally, assume X is proper. Let x ∈ X. Let (δ i j (x)) be an arbitrary subsequence of (δ i (x)). Since Examples 3.20 and 3.21 illustrate how the asserted equalities in the second parts of both Proposition 3.10(a,b) may fail. Notice that in Example 3.20 X is not a length space (and not proper) while in Example 3.21 X is not locally compact (and not proper).
It is useful to examine Proposition 3.10 in the setting when X is a length space; here we know that for any component Ω of an open U X, δ ∂Ω = δ ∂U = δ X\U in Ω. As in [3], a component Ω of U := core(U i ) is a limit region if and only if Ω is a component of core(U i j ) for every subsequence (U i j ) of (U i ). Adapting the ideas in the proof of Proposition 3.10 we deduce the following, which is part of the Quasihyperbolic Kernel Theorem in [3].

Let Ω be a component of U. Then Ω is a limit region for (U i ) if and only if δ
Here is a consequence of Theorem 3.1 in conjunction with Proposition 3.10(b).

Corollary. Let A, A
Ifdist H (A i , A) → , then (U i ) converges to its core U = core(U i ). Conversely, if (U i ) converges to its core U = core(U i ). Then: F is the only possibledist H -subsequential limit of (F i ) A, F are the only possibledist H -subsequential limits of (Â i )⁸.
Proof. Suppose there are E ∈ H(X) and a subsequence (F boundedly uniformly in X, where B := E ∩ X. Therefore Proposition 3.10(b) tells us that If E ∈ H(X) anddist H (A i j , E) → , we argue similarly but now it is possible that ∞ ̸ ∈ E and then Next we provide information relating core(U i ) and the core convergence of (U i ) to the existence of certain sequences (a i ) with a i ∈ A i .

Lemma. Let
(a) For each a ∈ A there is a sequence (a i ), with a i ∈ A i for all i, that subconverges to a. If (U i ) converges to U = core(U i ), then (a i ) converges to a. (b) All subsequential limits of any sequence (a i ), with a i ∈ A i for all i, lie in A.

. Kuratowksi Convergence
Let C be an in nite collection of non-empty subsets of a topological space X. Following Kuratowski and Whyburn (see [18], [28], [27]), we de nē It is not di cult to check that bothC and C are closed sets, and when B ⊂ C is in nite, C ⊂ B ⊂B ⊂C. When C = S =C, we say that C converges to S. We use similar notation and terminology for a sequence of non-empty subsets of X.
In particular, a sequence (A i ) of non-empty closed sets in X Kuratowski converges to a closed set A provided lim inf A i = A = lim sup A i . There is a direct connection with this notion of set convergence and that studied in §3.2. Proof. It is straightforward, using just de nitions, to verify (a). That Kuratowski convergence implies Carathéodory core convergence, as stated in (b), now follows. To complete (b), assume that (A i ) does not converge; so, there is a point x ∈ lim sup A i \ lim inf A i . This means that there is an open neighborhood V of x and i < i < . . . such that for all j, V ∩ A i j = ∅, or equivalently, V ⊂ U i j . Thus x ∈ core(U i j ) \ core(U i ). Now suppose that |·| is a compatible distance function on X, so the metric topology agrees with the original topology. Here are some alternative descriptions for lim inf A i , lim sup A i , and Kuratowski convergence. First, Also, we see that a sequence (A i ) of non-empty closed sets Kuratowski When X is proper, the following are equivalent: Proof. To see that (b) implies (a), note that as X is proper, every subsequence of (F i ) must have a further subsequence thatdist H -converges, and according to Corollary 3.12 its limit must be F. From Corollary 3.6, Proposition 3.8, Proposition 3.10(b), and Lemma 3.14 we obtain the following for proper spaces.
3.17 Corollary. For ∅ ≠ A, A i X closed with X proper, the following are equivalent. .

Convergence Examples
Here we collect some examples that illustrate the need for various hypotheses (such as properness, locally compact, (A i ) having UBP, etc.) required in many of our results. For the most part, proofs are left to the reader. First, we present a proper length space example.
, but now A and A ∪ {∞} are both subsequential limits of (Â i ) (and all the other properties continue to hold).
Next we o er two examples where X is a bounded, complete, locally compact, recti ably connected space   Here is an example of an unbounded complete length space that illustrates the need for (δ A i ) to converge boundedly uniformly in Theorem 3.5; it also reveals that in non-proper spaces, chordal Hausdor convergence and Kuratowski convergence are quite di erent. Keeping [25,Proposition 2.25] in mind we see that (z i ), (y i ), (x i ) are Gromov sequences in (Ω , k ) that represent the boundary points , ∞, e respectively. Since f i is an ε i -rough-isometry, we can readily check that the Gromov products satisfy , (x i ) are Gromov sequences in (Ω, k) that represent distinct boundary points. As this contradicts the fact that the Gromov boundary of (Ω, k) has cardinality two, we conclude that (Ω i , k i ; e) does not pointed Gromov-Hausdor converge to (Ω, k; e).
The moral here is that chordal Hausdor convergence is a handy tool, albeit somewhat technical. For analyzing complete conformal metrics in locally compact length spaces, Kuratowski convergence (and the equivalent core convergence) is more robust, easier to use, and provides more information (in the sense that we can verify pointed Gromov-Hausdor convergence, e.g., provided we have Kuratowski convergence, even when chordal Hausdor convergence fails). However, in non-proper spaces, Kuratowski convergence does not give pointed Gromov-Hausdor convergence whereas chordal Hausdor convergence does.
There is substantial interest in quasihyperbolic geometry in Banach spaces; e.g., Väisälä's de nition in [24] of a quasiconformal map between Banach space domains is based on quasihyperbolic distance. In this setting Kuratowski convergence cannot be employed, but chordal Hausdor convergence still gives useful results.

Convergence of Metrics
Now we turn our attention to conformal metrics and pointed Gromov-Hausdor convergence. We show that bounded uniform convergence of conformal metrics implies the same for their associated length distances; in particular, the associated length spaces pointed Gromov-Hausdor converge. As applications of these ideas, we get approximation results for: quasihyperbolic distance in domains in complete length spaces, Ferrand and Kulkarni-Pinkall-Thurston distances in quasihyperbolic domains inR n , and hyperbolic distance for hyperbolic domains inĈ. We conclude by examining Gromov-Hausdor convergence for Euclidean sets; especially, Corollary 5.9 characterizes pointed Gromov-Hausdor convergence of Euclidean quasihyperbolic spaces.

. Pointed Gromov-Hausdor Convergence
Here we prove that if ρ ds and ρ i ds are conformal metrics de ned in open subspaces U, U i ⊂ X of a quasihyperbolic superspace, and if (ρ i ds) converges to ρ ds in the appropriate sense, then the associated sequence (U i , dρ i ) of length spaces converges to (U, dρ) with respect to pointed Gromov-Hausdor convergence. The reader is no doubt familiar with the notion of local uniform convergence (aka uniform convergence on compact subsets) which is often a handy substitute for uniform convergence. Pointed Gromov-Hausdor convergence is speci cally designed to deal with the lack of compactness. As we saw in §3.1, compactness also plays no role in chordal Hausdor convergence; instead, convergence in bounded sets is the crucial ingredient.
Employing (4.6) we see that γ * is recti able as a path in the complete metric space (U, d), and as such it has an endpoint c := γ * (t * ) ∈ U with d(γ(t), c) → as t → t − * . Then d(a, c)   → U are well de ned, and are the desired ε-rough pointed isometries.
4.9 Remarks. The above argument consists of three parts. First, we have the "ball engul ng" property described in (4.5). Beardon and Minda illuminated the key ideas behind this fundamental property (in the complete, locally compact, length space setting); see [3]. Similar techniques have been useful in the study of BLD mappings (see, for example, [10] and [20]). Next, we have that bounded uniform convergence of the metrics gives "the same" for the associated length distances in the sense that (4.7) holds. Finally, we have the pointed Gromov-Hausdor convergence. In fact this last part is true in general: If (U i , d i ) are length spaces and (U, d) is a metric space such that for each a ∈ U, each R > , and each ε > there is an io with the property that for all

. Ferrand and Kulkarni-Pinkall-Thurston Convergence
Here we verify that in the Euclidean setting we have local uniform convergence (which implies bounded uniform convergence) of both Ferrand and Kulkarni-Pinkhall-Thurston metrics. Proof. Since rotations ofR n = S n are chordal isometries, as well as Ferrand and Kulkarni-Pinkall-Thurston isometries, we may assume that A contains the 'point at in nity'. Furthermore, we may assume that each A i contains at least two points, so each component ofR n \ A, and ofR n \ A i , is a quasihyperbolic domain. Let a ∈R n \ A and x r ∈ ( , dist(a, A)  ≤ log( + r) .
The middle equality above holds because J is an isometry of (R n+ + , h), and we have also used the well known estimate¹¹ h ≤ j that is valid in R n+ + ; here