1. Introduction

There are various notions of “boundaries at infinity’’ of metric spaces in the literature. One of these is the spherical boundary of certain unbounded metric spaces, as introduced in [1]. This is defined in detail in Section 2, but let us mention here that it is a byproduct of the concept of sphericalization, which replaces an unbounded metric by a conformally distorted bounded metric . This allows one to interpret results in [2] concerning the quasihyperbolizations of bounded length spaces in the context of certain unbounded spaces. To this end, the relationship between and , together with the relationship between their associated quasihyperbolizations, was studied in [1].

The spherical boundary is a key ingredient in studying the invertibility of the sphericalization process as is clear from the results in [1], but no detailed study of the links between features of and features of was carried out there. The current paper aims to throw more light on one such link by proving results of the following type: if the spherical boundary of a suitable metric measure space is sufficiently large, then has rapid volume growth. For instance, contrast Euclidean space whose spherical boundary is a singleton set (if ) with the hyperbolic plane whose spherical boundary has infinite Minkowski dimension.

After some preliminaries in Section 2, Section 3 examines annular convexity and conditions under which the spherical boundary is a singleton set and annular convexity, and Section 4 contains the main results.

2. Preliminaries

We denote by and the minimum and maximum, respectively, of numbers , .

2.1. Metric Spaces and Paths

Below, is always a metric space which may have additional properties as specified. We denote by the metric closure of and, viewing as a subset of , we write . Given denotes the class of rectifiable paths for which is a rectifiable path in , and . We also define to be the subset of consisting of paths that are parametrized by -arclength. We write or if the space needs to be specified.

Suppose is rectifiably connected and, only for this paragraph, let us write , . When restricted to , defines the inner metric associated with . We say that is a length metric, and that is a length space, if , for all ; this equality clearly extends to points . More generally, we say that a rectifiably connected metric space is a local length space if whenever , , and . is a geodesic space if, for all , there exists a path of length .

Every domain is a local length space when equipped with the Euclidean metric, and a slit disk in the Euclidean plane is a simple example of a local length space that is not a length space.

Given a local length space , we define the length boundary of , , to be the set of all points for which is nonempty for some (and hence all) . Equivalently is the set of all whose inner distance from some (and hence all) is finite. If is a length metric, then , but equality may fail if is merely a local length space. For instance, if is the Euclidean metric on a domain which spirals sufficiently tightly near some point , then .

The rest of our notation is quite standard. We denote by , , and , the open ball, closed ball, and sphere of radius about ; we omit the -subscript if the metric is understood. If , is the empty set. A metric space is proper if all its closed balls are compact.

An arc in is an injective path . We do not distinguish notationally between paths and their images. If is an arc in , and , is the subarc of with endpoints , . Given two -rectifiable arcs in metric spaces and with , we define the initial length map by the requirement that maps each initial segment of to the initial segment of that satisfies .

2.2. Metric Measure Spaces

A metric measure space is a metric space with an attached positive Borel measure which gives positive finite measure to all balls; if is a (local) length space we call a (local) length measure space.

Suppose is a metric measure space and . We say that is -doubling if whenever for fixed but arbitrary , . We say that is -translate doubling if instead whenever , are overlapping balls of the same radius or weak -translate doubling if we merely have whenever , are overlapping balls of radius .

A measure is doubling if and only if it is translate doubling and the underlying space has finite Assouad dimension (equivalently, all balls can be covered by a bounded number of balls of half the radius). Thus, doubling and translate doubling are (quantitatively) equivalent in Euclidean space and examples of length spaces with translate doubling measures that fail to be doubling including Hausdorff -measure on hyperbolic -space and arclength measure on the Cayley graph of an -generator free-group.

The lower Minkowski dimension of a subset of a metric space is defined by where is the maximum cardinality of a collection of disjoint open balls of radius and centers in .

2.3. The Spherical Boundary

A Borel function is said to be a -sphericalizing function, , if it has the following properties:(S1) whenever , and ;(S2).

We recall the following property of a sphericalizing function, taken from [1].

Lemma 2.1. If is a -sphericalizing function then(S3), for all .
In particular, as .

Suppose is an unbounded pointed local length space, and let us write , . Given a sphericalizing function , we define a new metric on by the equation: We usually write in place of .

If , , , then it is clear that is also of finite -length, so the length boundary can be viewed as a subset of . We define the spherical boundary of to be , and the spherical closure of to be . Since any point in is at a finite -distance from , and since is bounded away from zero on bounded intervals, it follows that a sequence in is -convergent to a point in if and only if it is -convergent. It also follows that if , , and , then cannot be contained in any ball .

We record some useful elementary estimates involving the two metrics and . Below, , , , and is the sphericalization constant of . It follows by standard analysis (as in [1, Proposition 2.14]) that , for all . Using (S2) and (S3), we readily deduce that Suppose for some , with , and let , where . Then , , and so using (S1) and (S2), we deduce that

3. Spherical Boundaries and Annular Convexity

A metric space is said to be -annular convex for some if for every , , and every pair of points there exists a path from to in the annulus of length at most .

Annular convex spaces, introduced by Herron et al. in [3], form a large class of spaces that include all Banach spaces (with the exception of one-dimensional real Banach spaces) and most spaces equipped with a doubling measure that supports a Poincaré inequality (as follows from the results in [4]).

In this section, we show that finite and countable products of metric spaces are annular convex. This is of interest to us because of the following simple result.

Proposition 3.1. The spherical boundary of an unbounded proper annular convex pointed length space is a singleton set.

Proof. Since is proper, it follows from [5, Theorem 2.4] that is nonempty. We write , , and let , where is a given -sphericalizing function.
Fixing a fix a pair of points , we pick sequences and in converging to and , respectively. Since necessarily and , we may assume that and , .
Joining to by a path of length at most , and picking a point on this path with the property that , it follows from (2.4) that . Using Lemma 2.1, we deduce that as . We similarly find points such that and as .
But by annular convexity and the properties of sphericalizing functions, it is easy to see that , where and so . Thus and so , as required.

Note that the assumption that is proper in Proposition 3.1 was needed only to show that the spherical boundary is nonempty. Some such condition is needed to deduce this fact: for instance if is a bouquet of line segments of length for each , joined together by identifying with each other the left endpoints of all such intervals, then it is easy to show that is empty.

It is clear from the proof of Proposition 3.1 that -annular convexity can be replaced by the following formally weaker condition: a metric space is weakly -annular convex, where , if for every , , and every pair of points , such that , there exists a path from to in of length at most . However, replacing annular convexity by weak annular convexity is of no real benefit in Proposition 3.1, since for length spaces the two conditions are quantitatively equivalent. We record the simple argument for completeness.

Proposition 3.2. If a length space is weak -annular convex, then it is -annular convex.

Proof. Consider distinct points . Let , . Join by a path of length less than , where is so small that . Clearly remains inside , so it certainly verifies the -annular convexity condition if it remains outside . Assume therefore that ventures inside this ball. Let , be the first and last points on such that , let be the initial segment of from to , let be the final segment of from to , and let be a path given by weak annular convexity for the pair , and center point . Let be the concatenation of , , and .
Then and weak annular convexity gives , so . The path intersects and its endpoints lie outside , so . Thus when is sufficiently small. By construction, remains outside and, since , it suffices to verify that also lies in this ball. But , and the endpoints of are a distance from , so as required.

It turns out that product spaces are annular convex.

Proposition 3.3. Let be the Cartesian product of unbounded length spaces and , with being the product of and . Then is a 4-annular convex length space.

We get the following immediate corollary of Propositions 3.1 and 3.3.

Corollary 3.4. Let be the Cartesian product of unbounded length spaces and , with being the product of and . Then the spherical boundary of is a singleton set.

Rather than proving Proposition 3.3, we prove a much more general result that is modeled on the previously mentioned fact that Banach spaces of dimension at least 2 are annular convex. We will generalize this to a large class of what could roughly be termed normed spaces with values in unbounded length spaces (which can vary from point to point). More precisely, we look at metrics constructed in the following manner.

We begin with a real normed vector space which we assume to consist of functions defined on an index set . We say that the norm on is monotonic on if(a) is a space of real-valued functions on , that is, .(b)If , , and for , then and .

Assume now that , are metric spaces and let . We define as follows the metric subproduct of , , relative to some fixed and the norm , which is monotonic on : if , then whenever defines a function , and we define , which makes sense by monotonicity of the norm. We write .

Theorem 3.5. Suppose is a normed vector space of dimension at least 2 which is monotonic on , and suppose , where each is an unbounded length space. Then is 4-annular convex.

Before proving Theorem 3.5, we discuss metric subproducts and give some examples. If , it is clear that , so certainly . It is routine to verify that is a metric on . Although is defined with respect to some , it is clear that we get the same metric space if is replaced by any . However, we get a metric subproduct disjoint from our original if we replace by any .

Lemma 3.6. Suppose is a normed vector space of dimension at least 2 which is monotonic on , and suppose is a length space for all . Then is also a length space.

Proof. Suppose and let be fixed but arbitrary. We define a path such that , with the following important properties which we record for later reference: In fact this is quite easy to do: since is a length space, we can certainly pick satisfying (3.3). Then for all and . Assembling together these paths to get a path , (3.4) follows from the last estimate and the fact that is defined via a monotone norm. In particular, (3.4) implies that . Since and are all arbitrary, the result follows.

Suppose we fix , where . For every and we can find a point such that : in fact the unboundedness of ensures that there exists such that , and then we use continuity to pick the required on a path from to . It follows that if , we can find a point such that and so .

We use function notation for versus subscript notation for to emphasize the difference between the normed space and the subproduct .

The simplest examples of monotonic normed spaces are spaces associated with finite or countably infinite , for . In the case of finite , the subproduct coincides as a set with the full Cartesian product . In the special case where has cardinality 2 and , we deduce Proposition 3.3.

Beyond the above spaces, other examples of monotonic-normed spaces include sums over uncountable index sets, but more interesting examples are normed sequence spaces of Orlicz or variable exponent type.

Note that if and each is the real line, then the subproduct is merely the normed space translated by a sequence : thus these subproducts are all cosets of , and any two such subproducts for different choices of can be put into a natural 1-1 correspondence.

However, there is not always such a natural 1-1 correspondence. Consider for instance the case where is the metric subspace of the real line given by for each and . If , then it is readily verified that the metric subproduct has the cardinality of the continuum, whereas if for any fixed , then is a countable space.

It can be shown that this dependence of the cardinality of on our choice of does not occur when the spaces , are length spaces (essentially because its cardinality is at least that of the continuum if is nontrivial). However, the above example suggests that there is in general no natural map from one metric subproduct to another.

Example 3.7. The constant 4 cannot be improved in Theorem 3.5. For instance if is the closed first quadrant of the -plane; this choice of corresponds to taking to be the plane, with being the Euclidean half line . Let , and where . Then every path from to intersects for any , as long as .

We now move on to the proof of Theorem 3.5.

Proof of Theorem 3.5. Let be the pair of points for which we want to verify the 4-annular convexity condition (with other data as usual), and let , so that . As in the proof of Proposition 3.2, a path connecting and of length at most , where , verifies the 4-annular convexity condition for data unless intersects . We may therefore assume that this intersection occurs and so Taking a limit as we get . In particular, .
Let us write for the characteristic function of any : thus if and only if . For and , let be defined by if , and if . For convenience, we write and for any .
Since has dimension at least 2, monotonicity readily implies that there are distinct indices such that the basic functions lie in .
We now define “scalar multiplication’’ on , restricted to scalar values . Choosing to be as in the proof of Lemma 3.6, with and , we let , where is the minimal such that . This definition is typically not unique since is not unique, but note that if then for all .
Suppose first that we can find some such as such that and , where . Let and for and , so that . Let , , and be the associated coordinate paths and argument for , as in the last paragraph.
Now join and by a path defined in the following piecewise manner by concatenating, in the natural order, paths , . First is a path from to which has component paths , where is a rescaled copy of if (and are as above), while is a constant speed path from to of length at most if . Thus, each is a constant speed path of length at most , and so .
Next let , for , and for . Then . Let be any path to such that the coordinate paths are stationary paths for and are of -length at most for . Since is a monotonic norm, we deduce that .
Finally and are analogues of and , respectively, but with replaced by , and with the directions of the paths reversed. It follows that the -length of our concatenated path is at most . Since and , we deduce that , as required.
We next need to show that , , stays outside . It follows from the definition of that for all , the same estimate follows for by monotonicity of the norm and symmetry with and then gives the same estimates for and , respectively.
Finally we need to show that each is contained in . The triangle inequality ensures that and for all . The same estimates for and follow by symmetry.
We may therefore make the added assumption that there is no way to split into complementary subsets and such that and . Note though that for any and , we either have or . By our added assumption, it follows that for every and , either and both hold for or and both hold for .
In particular, one of these last pairs of conditions holds for a set that contains but not . By switching the definitions of and if necessary, we assume that and both hold for , and that . We choose such that for and .
As in the previous case, we let for , but now we choose for . As before . Also let be the points satisfying and if , and .
We now join and by a path defined in the following piecewise manner by concatenating, in the natural order, paths , , where is a path from to defined as in the previous case, is a path of length at most from to which is stationary except in coordinate , is a path of length at most from to which is stationary in coordinate , is a path of length at most from to which is stationary except in coordinate , and is analogous to in reverse, but from to . As in the previous case, we see that . Since , by taking to be sufficiently small we get , as required. The fact that can be verified as before, so we leave it to the reader.

The examples that we have so far include the cases where is an or related space, but we cannot handle general spaces because the requirement that the norm is monotonic restricts us to spaces where nonnegative functions that are pointwise less than a given function in must also lie in . This is incompatible with spaces of measurable functions (unless the sigma algebra is the power set), let alone spaces of continuous or smooth functions. To get similar results for such spaces, the basic problem is getting fine control over the relationship between for different values of and fixed , where is as in Proposition 3.2. One way to get such control is to assume that each is a geodesic space, so that we can assume that is a constant speed geodesic. Then is also a constant speed geodesic and , allowing us to get analogues of Theorem 3.5 for more general spaces. The assumptions of monotonicity and dimension at least 2 would need to be replaced by assumptions appropriate to the context.