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ISRN Geometry
VolumeΒ 2012Β (2012), Article IDΒ 484312, 13 pages
http://dx.doi.org/10.5402/2012/484312
Research Article

The Spherical Boundary and Volume Growth

1Department of Mathematics, National University of Ireland Maynooth, Maynooth, Kildare, Ireland
2National Survey and Cadastre of Denmark, Rentemestervej 8, 2400 Copenhagen, Denmark

Received 29 November 2011; Accepted 3 January 2012

Academic Editors: F.Β Balibrea, J.Β Montaldi, and A.Β Morozov

Copyright Β© 2012 Stephen M. Buckley and Simon L. Kokkendorff. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the spherical boundary, a conformal boundary using a special class of conformal distortions. We prove that certain bounds on volume growth of suitable metric measure spaces imply that the spherical boundary is β€œsmall” (in cardinality or dimension) and give examples to show that the reverse implications fail. We also show that the spherical boundary of an annular convex proper length space consists of a single point. This result applies to 𝑙2-products of length spaces, since we prove that a natural metric, generalizing such β€œnorm-like” product metrics on a (possibly infinite) product of unbounded length spaces, is annular convex.

1. Introduction

There are various notions of β€œboundaries at infinity’’ of metric spaces in the literature. One of these is the spherical boundary πœ•π‘†π‘‹ 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 𝑙 by a conformally distorted bounded metric 𝜎. 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 𝑙 and 𝜎, together with the relationship between their associated quasihyperbolizations, was studied in [1].

The spherical boundary πœ•π‘†π‘‹ 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 𝑋 and features of πœ•π‘†π‘‹ 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 𝑋 has rapid volume growth. For instance, contrast Euclidean space ℝ𝑛 whose spherical boundary is a singleton set (if 𝑛>1) 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 π‘Žβˆ§π‘ and π‘Žβˆ¨π‘ the minimum and maximum, respectively, of numbers π‘Ž, 𝑏.

2.1. Metric Spaces and Paths

Below, (𝑋,𝑑) is always a metric space which may have additional properties as specified. We denote by 𝑋𝑑 the metric closure of (𝑋,𝑑) and, viewing 𝑋 as a subset of 𝑋𝑑, we write πœ•π‘‹π‘‘=𝑋𝑑⧡𝑋. Given π‘₯,π‘¦βˆˆπ‘‹π‘‘,Ξ“(π‘₯,𝑦) denotes the class of rectifiable paths πœ†βˆΆ[0,𝑇]→𝑋𝑑 for which πœ†|(0,𝑇) is a rectifiable path in 𝑋,πœ†(0)=π‘₯, and πœ†(𝑇)=𝑦. We also define Γ𝑑(π‘₯,𝑦) to be the subset of Ξ“(π‘₯,𝑦) consisting of paths that are parametrized by 𝑑-arclength. We write Ξ“(π‘₯,𝑦;𝑋) or Γ𝑑(π‘₯,𝑦;𝑋) if the space needs to be specified.

Suppose (𝑋,𝑑) is rectifiably connected and, only for this paragraph, let us write π‘‘ξ…ž(π‘₯,𝑦)=infπ›ΎβˆˆΞ“π‘‘(π‘₯,𝑦)len𝑑(𝛾), π‘₯,π‘¦βˆˆπ‘‹π‘‘. When restricted to 𝑋×𝑋, π‘‘ξ…ž defines the inner metric associated with 𝑑. We say that 𝑑 is a length metric, and that (𝑋,𝑑) is a length space, if 𝑑(π‘₯,𝑦)=π‘‘ξ…ž(π‘₯,𝑦), for all π‘₯,π‘¦βˆˆπ‘‹; this equality clearly extends to points π‘₯,π‘¦βˆˆπ‘‹π‘‘. More generally, we say that a rectifiably connected metric space (𝑋,𝑑) is a local length space if 𝑑(π‘₯,𝑦)=π‘‘ξ…ž(π‘₯,𝑦) whenever π‘₯βˆˆπ‘‹, π‘¦βˆˆπ‘‹π‘‘, and 𝑑(π‘₯,𝑦)≀𝑑(π‘₯,πœ•π‘‹π‘‘). (𝑋,𝑑) is a geodesic space if, for all π‘₯,π‘¦βˆˆπ‘‹, there exists a path π›ΎβˆˆΞ“π‘‘(π‘₯,𝑦) of length 𝑑(π‘₯,𝑦).

Every domain Ξ©βŠ‚β„π‘› 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 (𝑋,𝑑), we define the length boundary of (𝑋,𝑑), πœ•0𝑋𝑑, to be the set of all points π‘¦βˆˆπœ•π‘‹π‘‘ for which Γ𝑑(π‘₯,𝑦) is nonempty for some (and hence all) π‘₯βˆˆπ‘‹. Equivalently πœ•0𝑋𝑑 is the set of all π‘¦βˆˆπ‘‹π‘‘ whose inner distance from some (and hence all) π‘₯βˆˆπ‘‹ is finite. If 𝑑 is a length metric, then πœ•0𝑋𝑑=πœ•π‘‹π‘‘, but equality may fail if 𝑑 is merely a local length space. For instance, if 𝑑 is the Euclidean metric on a domain Ξ©βŠ‚β„π‘› which spirals sufficiently tightly near some point π‘¦βˆˆπœ•Ξ©π‘‘, then π‘¦βˆ‰πœ•0Ω𝑑.

The rest of our notation is quite standard. We denote by 𝐡𝑑(π‘₯,π‘Ÿ), 𝐡𝑑(π‘₯,π‘Ÿ), and 𝑆𝑑(π‘₯,π‘Ÿ), the open ball, closed ball, and sphere of radius π‘Ÿ about π‘₯βˆˆπ‘‹; we omit the 𝑑-subscript if the metric is understood. If π‘Ÿβ‰€0, 𝐡𝑑(π‘₯,π‘Ÿ) is the empty set. A metric space is proper if all its closed balls are compact.

An arc in 𝑋 is an injective path π›ΎβˆΆπΌβ†’π‘‹. We do not distinguish notationally between paths and their images. If 𝛾 is an arc in 𝑋, and 𝑒,π‘£βˆˆπ›Ύ, 𝛾[𝑒,𝑣] is the subarc of 𝛾 with endpoints 𝑒, 𝑣. Given two 𝑑-rectifiable arcs 𝛾,π›Ύξ…ž in metric spaces (𝑋,𝑑) and (π‘‹ξ…ž,π‘‘ξ…ž) with len𝑑(𝛾)≀len𝑑′(π›Ύξ…ž), we define the initial length map π‘“βˆΆπ›Ύβ†’π›Ύξ…ž by the requirement that 𝑓 maps each initial segment 𝛾[𝑒,𝑣] of 𝛾 to the initial segment π›Ύξ…ž[π‘’ξ…ž,π‘£ξ…ž] of π›Ύξ…ž that satisfies len𝑑′(π›Ύξ…ž[π‘’ξ…ž,π‘£ξ…ž])=len𝑑(𝛾[𝑒,𝑣]).

2.2. Metric Measure Spaces

A metric measure space (𝑋,𝑑,πœ‡) is a metric space with an attached positive Borel measure πœ‡ which gives positive finite measure to all balls; if (𝑋,𝑑) is a (local) length space we call (𝑋,𝑑,πœ‡) a (local) length measure space.

Suppose (𝑋,𝑑,πœ‡) is a metric measure space and 𝐢β‰₯1. We say that (𝑋,𝑑,πœ‡) is 𝐢-doubling if πœ‡(2𝐡)β‰€πΆπœ‡(𝐡) whenever 𝑑𝐡=𝐡(π‘₯,π‘‘π‘Ÿ) for fixed but arbitrary π‘₯βˆˆπ‘‹, π‘Ÿ>0. We say that (𝑋,𝑑,πœ‡) is 𝐢-translate doubling if instead πœ‡(π΅ξ…ž)β‰€πΆπœ‡(𝐡) whenever 𝐡, π΅ξ…ž are overlapping balls of the same radius or weak 𝐢-translate doubling if we merely have πœ‡(π΅ξ…ž)≀𝐢(1+π‘Ÿ)πœ‡(𝐡) whenever 𝐡, π΅ξ…ž are overlapping balls of radius π‘Ÿ.

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 𝑛-measure on hyperbolic 𝑛-space and arclength measure on the Cayley graph of an 𝑛-generator free-group.

The lower Minkowski dimension dim𝑀𝐸 of a subset 𝐸 of a metric space (𝑋,𝑑) is defined bydim𝑀(𝐸)=liminfπœ–β†’0+log𝑁(𝐸,πœ–),log(1/πœ–)(2.1) where 𝑁(𝐸,πœ–) is the maximum cardinality of a collection of disjoint open balls of radius πœ– and centers in 𝐸.

2.3. The Spherical Boundary

A Borel function π‘”βˆΆ[0,∞)β†’(0,∞) is said to be a 𝐢-sphericalizing function, 𝐢>2, if it has the following properties:(S1)𝑔(π‘Ÿ)≀𝐢𝑔(𝑠) whenever π‘Ÿ,𝑠β‰₯0,π‘Ÿβ‰€2𝑠+1, and 𝑠≀2π‘Ÿ+1;(S2)βˆ«βˆžπ‘Ÿπ‘”(𝑑)π‘‘π‘‘β‰€πΆπ‘Ÿπ‘”(π‘Ÿ),π‘Ÿβ‰₯1.

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

Lemma 2.1. If π‘”βˆΆ[0,∞)β†’(0,∞) is a 𝐢-sphericalizing function then(S3)𝑔(𝑠)/𝑔(π‘Ÿ)≀𝐢2(π‘Ÿ/𝑠)1+1/𝐢, for all 1β‰€π‘Ÿβ‰€π‘ .
In particular, 𝑑𝑔(𝑑)β†’0 as π‘‘β†’βˆž.

Suppose (𝑋,𝑙,π‘œ) is an unbounded pointed local length space, and let us write |π‘₯|=𝑙(π‘₯,π‘œ), π‘₯βˆˆπ‘‹π‘™. Given a sphericalizing function π‘”βˆΆ[0,∞)β†’(0,∞), we define a new metric 𝔖(𝑙,π‘œ,𝑔) on 𝑋 by the equation:𝔖(𝑙,π‘œ,𝑔)(π‘₯,𝑦)=infπ›ΎβˆˆΞ“(π‘₯,𝑦)ξ€œπ›Ύπ‘”(|𝑧|)𝑑𝑙(𝑧),π‘₯,π‘¦βˆˆπ‘‹.(2.2) We usually write 𝜎 in place of 𝔖(𝑙,π‘œ,𝑔).

If π›ΎβˆˆΞ“π‘™(π‘₯,𝑦), π‘₯βˆˆπ‘‹, π‘¦βˆˆπœ•π‘‹π‘™, then it is clear that 𝛾 is also of finite 𝜎-length, so the length boundary πœ•0𝑋𝑙 can be viewed as a subset of πœ•π‘‹πœŽ. We define the spherical boundary of 𝑋,πœ•S𝑋 to be πœ•π‘‹πœŽβ§΅πœ•0𝑋𝑙, and the spherical closure of 𝑋 to be π‘‹πœŽ. Since any point in π‘₯βˆˆπ‘‹βˆͺπœ•0𝑋𝑙 is at a finite 𝑙-distance from π‘œ, and since 𝑔 is bounded away from zero on bounded intervals, it follows that a sequence in 𝑋 is 𝜎-convergent to a point in 𝑋βˆͺπœ•0𝑋𝑙 if and only if it is 𝑙-convergent. It also follows that if π‘₯βˆˆπ‘‹, π‘¦βˆˆπœ•π‘†π‘‹, and π›ΎβˆˆΞ“πœŽ(π‘₯,𝑦), then 𝛾 cannot be contained in any ball 𝐡𝑙(π‘œ,π‘Ÿ).

We record some useful elementary estimates involving the two metrics 𝑙 and 𝜎=𝔖(𝑙,o,𝑔). Below, 𝐺(𝑑)≑(1+𝑑)𝑔(𝑑), |π‘₯|≑𝑙(π‘₯,π‘œ), π›Ώβˆž(π‘₯)=𝜎(π‘₯,πœ•π‘†(𝑋)), and 𝐢𝑔 is the sphericalization constant of 𝑔. It follows by standard analysis (as in [1, Proposition 2.14]) that π›Ώβˆžβˆ«(π‘₯)β‰₯∞|π‘₯|𝑔(𝑑)𝑑𝑑, for all π‘₯βˆˆπ‘‹. Using (S2) and (S3), we readily deduce thatπ›Ώβˆž(π‘₯)β‰₯πΆπ‘”βˆ’2𝐺(|π‘₯|),π‘₯βˆˆπ‘‹.(2.3) Suppose π›ΎβˆˆΞ“π‘™(π‘œ,π‘₯) for some π‘₯βˆˆπ‘‹, with len𝑙(𝛾)=𝐿<|π‘₯|+1, and let 𝜈=𝛾|[𝑠,𝐿], where 𝛾(𝑠)=𝑣,|𝑣|β‰₯1. Then |𝛾(𝑑)|∈(π‘‘βˆ’1,𝑑), 0≀𝑑≀𝐿, and so using (S1) and (S2), we deduce thatlenπœŽξ€œ(𝜈)=𝐿𝑠𝑔||||𝛾(𝑑)𝑑𝑑≀𝐢2π‘”ξ€œπΏ|𝑣|𝑔(𝑑)𝑑𝑑≀𝐢3𝑔𝐺(|𝑣|).(2.4)

3. Spherical Boundaries and Annular Convexity

A metric space (𝑋,𝑑) is said to be 𝐢-annular convex for some 𝐢β‰₯2 if for every π‘œβˆˆπ‘‹, π‘Ÿ>0, and every pair of points π‘₯1,π‘₯2∈𝐡(π‘œ,π‘Ÿ)⧡𝐡(π‘œ,π‘Ÿ/2) there exists a path 𝛾 from π‘₯1 to π‘₯2 in the annulus 𝐡(π‘œ,πΆπ‘Ÿ)⧡𝐡(π‘œ,π‘Ÿ/𝐢) of length at most 𝐢𝑑(π‘₯1,π‘₯2).

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 PoincarΓ© inequality (as follows from the results in [4]).

In this section, we show that finite and countable 𝑙2 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 (𝑋,𝑙,π‘œ) is a singleton set.

Proof. Since 𝑋 is proper, it follows from [5, Theorem 2.4] that πœ•π‘†π‘‹ is nonempty. We write |π‘Ž|=𝑙(π‘Ž,π‘œ), π‘Žβˆˆπ‘‹, and let 𝜎=𝔖(𝑙,π‘œ,𝑔), where 𝑔 is a given 𝐢𝑔-sphericalizing function.
Fixing a fix a pair of points 𝑧,π‘€βˆˆπœ•π‘†π‘‹, we pick sequences (𝑧𝑛) and (𝑀𝑛) in 𝑋 converging to 𝑧 and 𝑀, respectively. Since necessarily |𝑧𝑛|β†’βˆž and |𝑀𝑛|β†’βˆž, we may assume that |𝑧𝑛|>𝑛 and |𝑀𝑛|>𝑛, π‘›βˆˆβ„•.
Joining 𝑧𝑛 to π‘œ by a path of length at most |𝑧𝑛|+1, and picking a point π‘§ξ…žπ‘› on this path with the property that |π‘§ξ…žπ‘›|=𝑛, it follows from (2.4) that 𝜎(𝑧𝑛,π‘§ξ…žπ‘›)≀𝐢3𝑔𝑛𝑔(𝑛). Using Lemma 2.1, we deduce that 𝜎(𝑧𝑛,π‘§ξ…žπ‘›)β†’0 as π‘›β†’βˆž. We similarly find points π‘€ξ…žπ‘› such that |π‘€ξ…žπ‘›|=𝑛 and 𝜎(𝑀𝑛,π‘€ξ…žπ‘›)β†’0 as π‘›β†’βˆž.
But by annular convexity and the properties of sphericalizing functions, it is easy to see that 𝜎(π‘§ξ…žπ‘›,π‘€ξ…žπ‘›)β‰€πΆξ…žπ‘›π‘”(𝑛), where πΆξ…ž=πΆξ…ž(𝐢,𝐢𝑔) and so 𝜎(π‘§ξ…žπ‘›,π‘€ξ…žπ‘›)β†’0. Thus 𝜎(𝑧𝑛,𝑀𝑛)β†’0 and so 𝑧=𝑀, as required.

Note that the assumption that 𝑋 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 𝑋 is a bouquet of line segments of length 𝑛 for each π‘›βˆˆβ„•, joined together by identifying with each other the left endpoints of all such intervals, then it is easy to show that πœ•π‘†π‘‹ is empty.

It is clear from the proof of Proposition 3.1 that 𝐢-annular convexity can be replaced by the following formally weaker condition: a metric space (𝑋,𝑑) is weakly 𝐢-annular convex, where 𝐢β‰₯1, if for every π‘œβˆˆπ‘‹, π‘Ÿ>0, and every pair of points π‘₯1, π‘₯2βˆˆπ‘‹ such that 𝑑(π‘œ,π‘₯𝑖)=π‘Ÿ,𝑖=1,2, there exists a path 𝛾 from π‘₯1 to π‘₯2 in 𝑋⧡𝐡(π‘œ,π‘Ÿ/𝐢) of length at most 2πΆπ‘Ÿ. 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 𝐢-annular convex, then it is (3𝐢)-annular convex.

Proof. Consider distinct points π‘₯1,π‘₯2∈𝐡(π‘œ,π‘Ÿ)⧡𝐡(π‘œ,π‘Ÿ/2). Let π‘‘π‘–βˆΆ=𝑑(π‘œ,π‘₯𝑖), 𝑖=1,2. Join π‘₯1,π‘₯2 by a path πœ† of length less than 𝑑(π‘₯1,π‘₯2)+πœ–, where πœ–>0 is so small that 𝑑(π‘₯1,π‘₯2)+πœ–<2𝑑(π‘₯1,π‘₯2). Clearly πœ† remains inside 𝐡(π‘œ,3πΆπ‘Ÿ), so it certainly verifies the (3𝐢)-annular convexity condition if it remains outside 𝐡(π‘œ,π‘Ÿ/3). Assume therefore that πœ† ventures inside this ball. Let 𝑧1, 𝑧2 be the first and last points 𝑧 on πœ† such that 𝑑(𝑧,π‘œ)=π‘Ÿ/3, let 𝛾1 be the initial segment of πœ† from π‘₯1 to 𝑧1, let 𝛾2 be the final segment of πœ† from 𝑧2 to π‘₯2, and let 𝛾3 be a path given by weak annular convexity for the pair 𝑧1, 𝑧2 and center point π‘œ. Let 𝛾 be the concatenation of 𝛾1, 𝛾3, and 𝛾2.
Then len𝑑(𝛾1)+len𝑑(𝛾2)β‰€πœ–+𝑑(π‘₯1,π‘₯2) and weak annular convexity gives len𝑑(𝛾3)≀2πΆπ‘Ÿ/3, so len𝑑(𝛾)β‰€πœ–+𝑑(π‘₯1,π‘₯2)+2πΆπ‘Ÿ/3. The path πœ† intersects 𝐡(π‘œ,π‘Ÿ/3) and its endpoints lie outside 𝐡(π‘œ,π‘Ÿ/2), so π‘Ÿ/3=2(π‘Ÿ/2βˆ’π‘Ÿ/3)<𝑑(π‘₯1,π‘₯2)+πœ–. Thus len𝑑𝑑π‘₯(𝛾)<(2𝐢+1)1,π‘₯2ξ€Έξ€Έξ€·π‘₯+πœ–β‰€3𝐢𝑑1,π‘₯2ξ€Έ,(3.1) when πœ–>0 is sufficiently small. By construction, 𝛾 remains outside 𝐡(π‘œ,π‘Ÿ/3𝐢) and, since πœ†βŠ‚π΅(π‘œ,3πΆπ‘Ÿ), it suffices to verify that 𝛾3 also lies in this ball. But len(𝛾3)≀2πΆπ‘Ÿ/3, and the endpoints of 𝛾3 are a distance π‘Ÿ/3 from π‘œ, so 𝛾3βŠ‚π΅ξ‚΅π‘œ,(𝐢+1)π‘Ÿ3ξ‚ΆβŠ‚π΅(π‘œ,3πΆπ‘Ÿ),(3.2) as required.

It turns out that product spaces are annular convex.

Proposition 3.3. Let (𝑋,𝑑) be the Cartesian product of unbounded length spaces (𝑋1,𝑑1) and (𝑋2,𝑑2), with 𝑑 being the 𝑙2 product of 𝑑1 and 𝑑2. Then (𝑋,𝑑) is a 4-annular convex length space.

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

Corollary 3.4. Let (𝑋,𝑑) be the Cartesian product of unbounded length spaces (𝑋1,𝑑1) and (𝑋2,𝑑2), with 𝑑 being the 𝑙2 product of 𝑑1 and 𝑑2. Then the spherical boundary of (𝑋,𝑑) 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 (𝑉,β€–β‹…β€–) which we assume to consist of functions defined on an index set 𝐼. We say that the norm β€–β‹…β€– on 𝑉 is monotonic on 𝐼 if(a)𝑉 is a space of real-valued functions on 𝐼, that is, π‘‰βŠ†β„πΌ.(b)If π‘“βˆˆπ‘‰, π‘”βˆΆπΌβ†’β„, and |𝑔(𝑖)|≀|𝑓(𝑖)| for π‘–βˆˆπΌ, then π‘”βˆˆπ‘‰ and ‖𝑔‖≀‖𝑓‖.

Assume now that (𝑋𝑖,𝑑𝑖), π‘–βˆˆπΌ are metric spaces and let βˆπ‘ƒβˆΆ=π‘–βˆˆπΌπ‘‹π‘–. We define as follows the metric subproduct (𝑋,𝑑) of (𝑋𝑖,𝑑𝑖), π‘–βˆˆπΌ, relative to some fixed π‘Žβˆˆπ‘ƒ and the norm β€–β‹…β€–, which is monotonic on 𝐼: if π‘₯=(π‘₯𝑖)π‘–βˆˆπΌβˆˆπ‘ƒ, then π‘₯βˆˆπ‘‹ whenever 𝑓(𝑖)∢=𝑑𝑖(π‘Žπ‘–,π‘₯𝑖) defines a function π‘“βˆˆπ‘‰, and we define 𝑑(π‘₯,𝑦)=‖𝑖↦𝑑𝑖(π‘₯𝑖,𝑦𝑖)β€–, which makes sense by monotonicity of the norm. We write ∏(𝑋,𝑑)=𝑉;π‘Žπ‘–βˆˆπΌ(𝑋𝑖,𝑑𝑖).

Theorem 3.5. Suppose 𝑉 is a normed vector space of dimension at least 2 which is monotonic on 𝐼, and suppose ∏(𝑋,𝑑)=𝑉;π‘Žπ‘–βˆˆπΌ(𝑋𝑖,𝑑𝑖), where each (𝑋𝑖,𝑑𝑖) is an unbounded length space. Then (𝑋,𝑑) is 4-annular convex.

Before proving Theorem 3.5, we discuss metric subproducts 𝑋 and give some examples. If π‘₯,π‘₯ξ…žβˆˆπ‘‹, it is clear that 𝑑(π‘₯,π‘₯ξ…ž)≀𝑑(π‘₯,π‘Ž)+𝑑(π‘Ž,π‘₯ξ…ž), so certainly 𝑑(π‘₯,π‘₯ξ…ž)<∞. It is routine to verify that 𝑑 is a metric on 𝑋. Although 𝑋 is defined with respect to some π‘Žβˆˆπ‘ƒ, it is clear that we get the same metric space (𝑋,𝑑) if π‘Ž is replaced by any π‘βˆˆπ‘‹. However, we get a metric subproduct disjoint from our original 𝑋 if we replace π‘Ž by any π‘βˆˆπ‘ƒβ§΅π‘‹.

Lemma 3.6. Suppose 𝑉 is a normed vector space of dimension at least 2 which is monotonic on 𝐼, and suppose (𝑋𝑖,𝑑𝑖) is a length space for all π‘–βˆˆπΌ. Then ∏(𝑋,𝑑)=𝑉;π‘Žπ‘–βˆˆπΌ(𝑋𝑖,𝑑𝑖) is also a length space.

Proof. Suppose π‘₯,π‘₯ξ…žβˆˆπ‘‹ and let πœ–>0 be fixed but arbitrary. We define a path πœ†βˆΆ[0,1]→𝑋 such that πœ†(𝑑)=(πœ†π‘–(𝑑))π‘–βˆˆπΌ,0≀𝑑≀1, with the following important properties which we record for later reference: πœ†π‘–βˆΆ[]0,1βŸΆπ‘‹π‘–isaconstantspeedpathfromπ‘₯𝑖toπ‘₯ξ…žπ‘–,π‘–βˆˆπΌ,lenπ‘‘ξ€·πœ†π‘–ξ€Έβ‰€(1+πœ–)𝑑𝑖π‘₯𝑖,π‘₯ξ…žπ‘–ξ€Έξ€·,π‘–βˆˆπΌ,(3.3)𝑑(πœ†(𝑠),πœ†(𝑑))≀(1+πœ–)(π‘‘βˆ’π‘ )𝑑π‘₯,π‘₯ξ…žξ€Έ,0≀𝑠≀𝑑≀1.(3.4) In fact this is quite easy to do: since 𝑋𝑖 is a length space, we can certainly pick πœ†π‘– satisfying (3.3). Then π‘‘π‘–ξ€·πœ†π‘–(𝑠),πœ†π‘–ξ€Έ(𝑑)≀lenπ‘‘π‘–ξ‚€πœ†π‘–||[𝑠,𝑑]=(π‘‘βˆ’π‘ )lenπ‘‘π‘–ξ€·πœ†π‘–ξ€Έβ‰€(1+πœ–)(π‘‘βˆ’π‘ )𝑑𝑖π‘₯ξ…žπ‘–,π‘¦ξ…žπ‘–ξ€Έ,(3.5) for all π‘–βˆˆπΌ and 0≀𝑠≀𝑑≀1. Assembling together these paths πœ†π‘– to get a path πœ†βˆΆ[0,1]→𝑋, (3.4) follows from the last estimate and the fact that 𝑑 is defined via a monotone norm. In particular, (3.4) implies that len𝑑(πœ†)≀(1+πœ–)𝑑(π‘₯,π‘₯β€²). Since π‘₯,π‘₯β€²βˆˆπ‘‹ and πœ–>0 are all arbitrary, the result follows.

Suppose we fix π‘œβˆˆπ‘‹, where π‘œ=(π‘œπ‘–)π‘–βˆˆπΌ. For every π‘–βˆˆπΌ and 𝑅>0 we can find a point 𝑧𝑖(𝑅)βˆˆπ‘‹π‘– such that 𝑑𝑖(π‘œπ‘–,𝑧𝑖(𝑅))=𝑅: in fact the unboundedness of 𝑋𝑖 ensures that there exists π‘§βˆˆπ‘‹π‘– such that 𝑑𝑖(π‘œπ‘–,𝑧)β‰₯𝑅, and then we use continuity to pick the required 𝑧𝑖(𝑅) on a path from 𝑧 to π‘œπ‘–. It follows that if π‘£βˆˆπ‘‰, we can find a point π‘₯𝑣=(π‘₯𝑣𝑖)π‘–βˆˆπΌβˆˆπ‘‹ such that 𝑑𝑖(π‘₯𝑣𝑖,π‘œπ‘–)=|𝑣𝑖| and so 𝑑(π‘₯𝑣,π‘œ)=‖𝑣‖.

We use function notation 𝑓(𝑖) for 𝑉 versus subscript notation π‘₯𝑖 for 𝑋 to emphasize the difference between the normed space (𝑉,β€–β‹…β€–) and the subproduct (𝑋,𝑑).

The simplest examples of monotonic normed spaces are 𝑙𝑝 spaces associated with finite or countably infinite 𝐼, for 1β‰€π‘β‰€βˆž. In the case of finite 𝐼, the subproduct 𝑋 coincides as a set with the full Cartesian product βˆπ‘–βˆˆπΌπ‘‹π‘–. In the special case where 𝐼 has cardinality 2 and 𝑝=2, we deduce Proposition 3.3.

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

Note that if 𝐼=β„• and each 𝑋𝑖 is the real line, then the subproduct is merely the normed space 𝑉 translated by a sequence π‘Žβˆˆβ„β„•: thus these subproducts are all cosets of 𝑉, and any two such subproducts for different choices of π‘Ž 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 𝑋𝑖 is the metric subspace of the real line given by 𝑋𝑖⋃={0}βˆͺ(π‘›βˆˆβ„•{𝑛,1/𝑛}) for each π‘–βˆˆβ„• and 𝑉=𝑙2. If π‘Ž=(0)π‘–βˆˆβ„•, then it is readily verified that the metric subproduct βˆπ‘‹βˆΆ=𝑉;π‘Žπ‘›βˆˆβ„•π‘‹π‘– has the cardinality of the continuum, whereas if π‘Ž=(𝑐)π‘–βˆˆβ„• for any fixed π‘βˆˆπ‘‹π‘–β§΅{0}, then 𝑋 is a countable space.

It can be shown that this dependence of the cardinality of 𝑋 on our choice of π‘Ž does not occur when the spaces (𝑋𝑖,𝑑𝑖), π‘–βˆˆβ„• are length spaces (essentially because its cardinality is at least that of the continuum if 𝑉 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 𝑋 is the closed first quadrant of the 𝑙1-plane; this choice of 𝑋 corresponds to taking 𝑉 to be the 𝑙1 plane, with 𝑋1=𝑋2 being the Euclidean half line [0,∞). Let π‘Ÿ=4,π‘₯=(0,0),𝑦=(2,0), and π‘œ=(1+𝛿,1+𝛿) where 0<𝛿<1. Then every path from π‘₯ to 𝑦 intersects 𝐡(π‘œ,π‘π‘Ÿ) for any 𝑐>1/4, as long as 𝛿<4π‘βˆ’1.

We now move on to the proof of Theorem 3.5.

Proof of Theorem 3.5. Let π‘₯,π‘¦βˆˆπ΅(π‘œ,π‘Ÿ)⧡𝐡(π‘œ,π‘Ÿ/2) be the pair of points for which we want to verify the 4-annular convexity condition (with other data π‘Ÿ,π‘œ as usual), and let π‘†βˆΆ=𝑑(π‘₯,π‘œ)+𝑑(𝑦,π‘œ), so that 𝑆>π‘Ÿ. As in the proof of Proposition 3.2, a path 𝛾 connecting π‘₯ and 𝑦 of length at most (1+πœ–)𝑑(π‘₯,𝑦), where πœ–>0, verifies the 4-annular convexity condition for data (π‘₯,𝑦,π‘œ,π‘Ÿ) unless 𝛾 intersects 𝐡(π‘œ,π‘Ÿ/4). We may therefore assume that this intersection occurs and so (1+πœ–)𝑑(π‘₯,𝑦)β‰₯lenπ‘‘ξ‚€π‘Ÿ(𝛾)>𝑑(π‘₯,π‘œ)βˆ’4+ξ‚€π‘Ÿπ‘‘(𝑦,π‘œ)βˆ’4.(3.6) Taking a limit as πœ–β†’0 we get 𝑑(π‘₯,𝑦)β‰₯π‘†βˆ’π‘Ÿ/2. In particular, 𝑑(π‘₯,𝑦)>𝑆/2.
Let us write π‘’π΄βˆˆ{0,1}𝐼 for the characteristic function of any π΄βŠ‚πΌ: thus 𝑒𝐴(𝑖)=1 if and only if π‘–βˆˆπ΄. For π‘₯βˆˆπ‘‹ and π΄βŠ‚πΌ, let π‘₯π΄βˆˆπ‘‹ be defined by π‘₯𝐴𝑖=π‘₯𝑖 if π‘–βˆˆπ΄, and π‘₯𝐴𝑖=π‘œπ‘– if π‘–βˆ‰π΄. For convenience, we write π‘’π‘–βˆΆ=𝑒{𝑖} and π‘₯π‘–βˆΆ=π‘₯{𝑖} for any π‘–βˆˆπΌ.
Since 𝑉 has dimension at least 2, monotonicity readily implies that there are distinct indices 𝑗,π‘˜βˆˆπΌ such that the basic functions 𝑒𝑗,π‘’π‘˜ lie in 𝑉.
We now define β€œscalar multiplication’’ on 𝑋, restricted to scalar values 0≀𝛼≀1. Choosing πœ†βˆΆ[0,1]→𝑋 to be as in the proof of Lemma 3.6, with (π‘₯,π‘₯ξ…ž,πœ–)=(π‘₯,π‘œ,πœ–) and 0<πœ–β‰€1/3, we let 𝛼⋅π‘₯∢=πœ†(𝑠), where 𝑠 is the minimal π‘ β€²βˆˆ[0,1] such that 𝑑(πœ†(π‘ ξ…ž),π‘œ)=𝛼𝑑(π‘₯,π‘œ). This definition is typically not unique since πœ† is not unique, but note that if π‘₯𝑖=π‘œπ‘– then (𝛼⋅π‘₯)𝑖=π‘œπ‘– for all 0≀𝛼≀1.
Suppose first that we can find some such as 𝐴 such that 𝑑(π‘₯𝐴,π‘œ)β‰₯π‘Ÿ/4 and 𝑑(𝑦𝐡,π‘œ)β‰₯π‘Ÿ/4, where 𝐡∢=𝐼⧡𝐴. Let 𝑧π‘₯∢=𝛼⋅π‘₯𝐴 and π‘§π‘¦βˆΆ=𝛽⋅𝑦𝐡 for 𝛼=π‘Ÿ/4𝑑(π‘₯𝐴,π‘œ) and 𝛽=π‘Ÿ/4𝑑(𝑦𝐡,π‘œ), so that 𝑑(𝑧π‘₯,π‘œ)=𝑑(𝑧𝑦,π‘œ)=π‘Ÿ/4. Let πœ†π‘–, π‘–βˆˆπ΄, and 𝑠 be the associated coordinate paths and argument for 𝛼⋅π‘₯𝐴, as in the last paragraph.
Now join π‘₯ and 𝑦 by a path 𝛾 defined in the following piecewise manner by concatenating, in the natural order, paths π›Ύπ‘šβˆΆ[0,1]→𝑋, 1β‰€π‘šβ‰€4. First 𝛾1 is a path from π‘₯ to 𝑧π‘₯ which has component paths 𝛾1π‘–βˆΆ[0,1]→𝑋𝑖, where 𝛾1𝑖 is a rescaled copy of πœ†π‘–|[0,𝑠] if π‘–βˆˆπ΄ (and πœ†=(πœ†π‘–),𝑠 are as above), while 𝛾1𝑖 is a constant speed path from π‘₯𝑖 to π‘œπ‘– of length at most (1+πœ–)𝑑𝑖(π‘₯𝑖,π‘œπ‘–) if π‘–βˆˆπ΅. Thus, each 𝛾𝑖 is a constant speed path of length at most (1+πœ–)𝑑𝑖(π‘₯𝑖,π‘œπ‘–), and so len𝑑(𝛾1)≀(1+πœ–)𝑑(π‘₯,π‘œ).
Next let 𝑧π‘₯𝑦=(𝑧𝑖π‘₯𝑦)π‘–βˆˆπΌ, 𝑧𝑖π‘₯𝑦=𝑧π‘₯𝑖 for π‘–βˆˆπ΄, and 𝑧𝑖=𝑧𝑦𝑖 for π‘–βˆˆπ΅. Then 𝑑(𝑧π‘₯,𝑧π‘₯𝑦)=𝑑(𝑧𝑦,𝑧π‘₯𝑦)=π‘Ÿ/4. Let 𝛾2∢[0,1]→𝑋 be any path 𝑧π‘₯ to 𝑧 such that the coordinate paths 𝛾2𝑖 are stationary paths for π‘–βˆˆπ΄ and are of 𝑑𝑖-length at most (1+πœ–)𝑑𝑖(𝑧π‘₯𝑖,𝑧𝑖π‘₯𝑦) for π‘–βˆˆπ΅. Since β€–β‹…β€– is a monotonic norm, we deduce that len𝑑(𝛾2)≀(1+πœ–)π‘Ÿ/4.
Finally 𝛾3 and 𝛾4 are analogues of 𝛾2 and 𝛾1, respectively, but with (π‘₯,𝐴,𝐡) replaced by (𝑦,𝐡,𝐴), and with the directions of the paths reversed. It follows that the 𝑑-length of our concatenated path 𝛾 is at most (1+πœ–)(𝑆+π‘Ÿ/2)≀3(1+πœ–)𝑆/2. Since 𝑑(π‘₯,𝑦)β‰₯𝑆/2 and πœ–β‰€1/3, we deduce that len(𝛾)≀4𝑑(π‘₯,𝑦), as required.
We next need to show that π›Ύπ‘š, 1β‰€π‘šβ‰€4, stays outside 𝐡(π‘œ,π‘Ÿ/4). It follows from the definition of 𝛼⋅π‘₯ that 𝑑(πœ†1(𝑑),π‘œ)β‰₯π‘Ÿ/4 for all 0≀𝑑≀1, the same estimate follows for 𝛾2 by monotonicity of the norm and symmetry with 𝛾2 and 𝛾1 then gives the same estimates for 𝛾3 and 𝛾4, respectively.
Finally we need to show that each π›Ύπ‘š is contained in 𝐡(π‘œ,4π‘Ÿ). The triangle inequality ensures that 𝑑(𝛾1(𝑑),π‘œ)<π‘Ÿ+(πœ–/2)π‘Ÿ<4π‘Ÿ and 𝑑(𝛾2(𝑑),π‘œ)<(π‘Ÿ/4)+(1+πœ–)(π‘Ÿ/4)<4π‘Ÿ for all 0≀𝑑≀1. The same estimates for 𝛾3 and 𝛾4 follow by symmetry.
We may therefore make the added assumption that there is no way to split 𝐼 into complementary subsets 𝐴 and 𝐡 such that 𝑑(π‘₯𝐴,π‘œ)β‰₯π‘Ÿ/4 and 𝑑(𝑦𝐡,π‘œ)β‰₯π‘Ÿ/4. Note though that for any π‘€βˆˆπ‘‹ and π΄βŠ‚πΌ, we either have 𝑑(𝑀𝐴,π‘œ)β‰₯𝑑(𝑀,π‘œ)βˆ’π‘Ÿ/4>π‘Ÿ/4 or 𝑑(𝑀𝐼⧡𝐴,π‘œ)β‰₯π‘Ÿ/4. By our added assumption, it follows that for every π΄βŠ‚πΌ and 𝐡∢=𝐼⧡𝐴, either 𝑑(𝑀𝐴,π‘œ)β‰₯𝑑(𝑀,π‘œ)βˆ’π‘Ÿ/4 and 𝑑(𝑀𝐡,π‘œ)<π‘Ÿ/4 both hold for π‘€βˆˆ{π‘₯,𝑦} or 𝑑(𝑀𝐡,π‘œ)β‰₯𝑑(𝑀,π‘œ)βˆ’π‘Ÿ/4 and 𝑑(𝑀𝐴,π‘œ)<π‘Ÿ/4 both hold for π‘€βˆˆ{π‘₯,𝑦}.
In particular, one of these last pairs of conditions holds for a set 𝐴 that contains 𝑗 but not π‘˜. By switching the definitions of (𝐴,𝑗) and (𝐡,π‘˜) if necessary, we assume that 𝑑(𝑀𝐴,π‘œ)β‰₯𝑑(𝑀,π‘œ)βˆ’π‘Ÿ/4 and 𝑑(𝑀𝐡,π‘œ)<π‘Ÿ/4 both hold for π‘€βˆˆ{π‘₯,𝑦}, and that π‘—βˆˆπ΄. We choose π‘€βˆˆπ‘‹ such that 𝑀𝑖=π‘œπ‘– for π‘–β‰ π‘˜ and 𝑑(𝑀,π‘œ)=π‘Ÿ/4.
As in the previous case, we let 𝑧π‘₯∢=𝛼⋅π‘₯𝐴 for 𝛼=π‘Ÿ/4𝑑(π‘₯𝐴,π‘œ), but now we choose π‘§π‘¦βˆΆ=𝛽⋅𝑦𝐴 for 𝛽=π‘Ÿ/4𝑑(𝑦𝐴,π‘œ). As before 𝑑(𝑧π‘₯,π‘œ)=𝑑(𝑧𝑦,π‘œ)=π‘Ÿ/4. Also let 𝑀π‘₯,𝑀𝑦 be the points satisfying 𝑀π‘₯𝑖=𝑧π‘₯𝑖 and 𝑀𝑦𝑖=𝑧𝑦𝑖 if π‘–β‰ π‘˜, and 𝑀π‘₯π‘˜=π‘€π‘¦π‘˜=π‘€π‘˜.
We now join π‘₯ and 𝑦 by a path 𝛾 defined in the following piecewise manner by concatenating, in the natural order, paths π›Ύπ‘šβˆΆ[0,1]→𝑋, 1β‰€π‘šβ‰€5, where 𝛾1 is a path from π‘₯ to 𝑧π‘₯ defined as in the previous case, 𝛾2 is a path of length at most (1+πœ–)π‘Ÿ/4 from 𝑧π‘₯ to 𝑀π‘₯ which is stationary except in coordinate π‘˜, 𝛾3 is a path of length at most (1+πœ–)π‘Ÿ/2 from 𝑀π‘₯ to 𝑀𝑦 which is stationary in coordinate π‘˜, 𝛾4 is a path of length at most (1+πœ–)π‘Ÿ/4 from 𝑀𝑦 to 𝑧𝑦 which is stationary except in coordinate π‘˜, and 𝛾5 is analogous to 𝛾1 in reverse, but from 𝑧𝑦 to 𝑦. As in the previous case, we see that len(𝛾)≀(1+πœ–)𝑆+π‘Ÿ. Since 𝑑(π‘₯,𝑦)>𝑆/2>π‘Ÿ/2, by taking πœ– to be sufficiently small we get len(𝛾)≀4𝑑(π‘₯,𝑦), as required. The fact that π›ΎβŠ‚π΅(π‘œ,4π‘Ÿ)⧡𝐡(π‘œ,π‘Ÿ/4) can be verified as before, so we leave it to the reader.

The examples that we have so far include the cases where 𝑉 is an 𝑙𝑝 or related space, but we cannot handle general 𝐿𝑝 spaces because the requirement that the norm is monotonic restricts us to spaces 𝑉 where nonnegative functions that are pointwise less than a given function in 𝑉 must also lie in 𝑉. 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 𝑑𝑖(πœ†π‘–(𝑑),π‘œπ‘–) for different values of π‘–βˆˆπΌ and fixed 0<𝑑<1, where πœ†=(πœ†π‘–) is as in Proposition 3.2. One way to get such control is to assume that each (𝑋𝑖,𝑑𝑖) is a geodesic space, so that we can assume that πœ†π‘–βˆΆ[0,1]→𝑋𝑖 is a constant speed geodesic. Then πœ†=(πœ†π‘–) is also a constant speed geodesic and 𝑑(πœ†(𝑑),π‘œ)=(1βˆ’π‘‘)𝑑(π‘₯,π‘œ), 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.

4. Large Spherical Boundary and Fast Volume Growth

A metric measure space, even a proper one, can have very fast volume growth and small spherical boundary, in the sense that its spherical boundary is a singleton set. For instance the product Riemannian manifold 𝑋=𝐻2×𝐻2 has exponential volume growth and constant negative Ricci curvature, but Corollary 3.4 implies that πœ•π‘†π‘‹ is a one-point space.

However, implications in the reverse direction are possible. Our Guiding Principle is that for reasonably general classes of pointed length measure spaces (𝑋,𝑙,π‘œ,πœ‡), a large spherical boundary forces (𝑋,𝑙) to have rapid volume growth. By making appropriate choices for the vague italicized phrases in our Guiding Principle, we get some theorems. We state and prove three such results in this section and discuss some relevant examples. In all instances, the reasonably large class of spaces consists of spaces satisfying a doubling condition or some weak variant thereof.

Throughout this section, (𝑋,𝑙,π‘œ) is a pointed length space, 𝑔 is a 𝐢𝑔-sphericalizing function, with associated spherical metric 𝜎=𝔖(𝑙,π‘œ,𝑔) and spherical boundary πœ•π‘†π‘‹. Also 𝐺(𝑑)=(1+𝑑)𝑔(𝑑),|π‘₯|=𝑙(π‘₯,π‘œ), and π›Ώβˆž(π‘₯)=dist𝜎(π‘₯,πœ•π‘†π‘‹).

In our first result, we assume that our metric measure space is doubling. This is a rather strong condition and it implies slow (meaning polynomial rate) volume growth so, without any explicit mention of volume growth, we deduce that the spherical boundary is quite small in the sense of having finite cardinality.

Theorem 4.1. Suppose (𝑋,𝑙,πœ‡) is πΆπœ‡-doubling. Then πœ•π‘†π‘‹ is a finite set whose cardinality is bounded by a number dependent only on 𝐢𝑔 and πΆπœ‡.

In our other results, we replace doubling by translate doubling or weak translate doubling. Unlike doubling, (weak) translate doubling puts no real constraint on volume growth, so volume growth enters the statements of our results explicitly.

Theorem 4.2. Suppose (𝑋,𝑙,πœ‡) is πΆπœ‡-translate doubling. If dimπ‘€πœ•π‘†π‘‹>0, then 𝑓(π‘Ÿ)β‰‘πœ‡(𝐡𝑙(π‘œ,π‘Ÿ)) grows faster than any polynomial. In fact, liminfπ‘Ÿβ†’βˆžlog𝑓(π‘Ÿ)log2π‘Ÿ>0.(4.1)

Theorem 4.3. Suppose (𝑋,𝑙,πœ‡) is weak πΆπœ‡-translate doubling. If dimπ‘€πœ•π‘†π‘‹>0, then 𝑓(π‘Ÿ)∢=πœ‡(𝐡𝑙(π‘œ,π‘Ÿ)) grows at a polynomial rate or faster, that is, liminfπ‘Ÿβ†’βˆžlog𝑓(π‘Ÿ)logπ‘Ÿ>0.(4.2)

Proof of Theorem 4.1. Suppose πœ•π‘†π‘‹ has at least 𝑁 points 𝑧1,…,𝑧𝑁. Choose 0<πœ–<(π›Ώβˆž(π‘œ)/5)∧(𝐺(1)/9𝐢3𝑔) so small that the balls 4π΅π‘–β‰‘π΅πœŽ(𝑧𝑖,4πœ–) are all disjoint and choose points π‘’π‘–βˆˆπ΅π‘–.
Suppose π›Ώβˆž(𝑧)≀3πœ–. Using (2.3), we get 𝐺(|𝑧|)𝐢2𝑔≀𝐺(1)3𝐢3𝑔.(4.3) It now follows from (S1) and the definition of 𝐺 that |𝑧|β‰₯2.
We carry out the following construction for each index 𝑖, 1≀𝑖≀𝑁. Choose π›Ύπ‘–βˆˆΞ“π‘™(π‘œ,𝑒𝑖) with len𝑙(𝛾)=𝐿<|𝑒𝑖|+1, and let π΅ξ…žπ‘–βˆΆ=𝐡𝜎(𝑣𝑖,πœ–), where 𝑣𝑖 is the first point at which 𝛾𝑖 meets 3𝐡𝑖. Now π‘‘βˆž(𝑣𝑖)≀3πœ–, and so |𝑣𝑖|β‰₯2. Using (2.3) and (2.4), we see that 𝐺(|𝑣𝑖|)β‰ˆπœ–. In view of (S3), we see that the distances |𝑣𝑖| are mutually comparable, so let us choose a pair of mutually comparable radii π‘Ÿ,𝑅 such that 2β‰€π‘Ÿβ‰€|𝑣𝑖|≀𝑅. By (S1), 𝐡𝑙(𝑣𝑖,𝑑)βŠ‚π΅πœŽ(𝑣𝑖,𝐢𝑔𝑑𝑔(𝑑)) for every 0<π‘‘β‰€π‘Ÿ. We can therefore fix π‘‘β‰€π‘Ÿ,π‘‘β‰ˆπ‘Ÿ, so that π΅π‘–ξ…žξ…žβˆΆ=𝐡𝑙(𝑣𝑖,𝑑)βŠ‚π΅ξ…žπ‘–.
Every π΅π‘–ξ…žξ…ž is contained in the single ball 𝐡0=𝐡𝑙(π‘œ,𝑅+𝑑) and in turn 𝐡0 is contained in each of the balls π‘ π΅π‘–ξ…žξ…ž, 𝑠=(2𝑅+𝑑)/𝑑. Since 𝑑 and 𝑅 are comparable, doubling ensures that πœ‡(𝐡0)≀𝐢1πœ‡(π΅π‘–ξ…žξ…ž), where 𝐢1 depends only on πΆπœ‡ and (2𝑅+𝑑)/𝑑≲1. Since 𝐡0 contains 𝑁 disjoint balls of measure at least πœ‡(𝐡0)/𝐢1, it follows that 𝑁≀𝐢1, as required.

Proof of Theorem 4.2. Part of the proof is similar to that of Theorem 4.1, so we will be sketchy. Since dimπ‘€πœ•π‘†π‘‹>0, there are constants 𝑐,𝑄>0 such that πœ•π‘†π‘‹ contains π‘πœ–βˆ’π‘„ disjoint 𝜎-balls π΅πœ–,𝑖 of radius πœ– for all 0<πœ–β‰€dia𝜎(𝑋). Taking πœ–π‘—=π΄βˆ’π‘—dia𝜎(𝑋) for a fixed number 𝐴>1, we can associate radii 𝑑𝑗 such that each of the π‘πœ–π‘—βˆ’π‘„ balls π΅πœ–π‘—,𝑖 contains an 𝑙-ball π΅ξ…žξ…žπ‘—,𝑖 whose radius is 𝑑𝑗 and whose distance from the origin is contained in the interval [π‘Ÿπ‘—,𝑅𝑗] for some numbers π‘Ÿπ‘—, 𝑅𝑗 are comparable with 𝑑𝑗; the constants of comparability can be taken to depend only 𝐢𝑔. We assume, as we may, that 𝐴 is chosen so large that π‘πœ–π‘—βˆ’π‘„β‰₯2𝑗, π‘Ÿ1β‰₯1, and 𝑅𝑗+2𝑑𝑗<π‘Ÿπ‘—+1, π‘—βˆˆβ„•. Note also that the ratios π‘Ÿπ‘—+1/π‘Ÿπ‘— are uniformly bounded by a constant dependent only on 𝐴 and 𝐢𝑔, so that logπ‘Ÿπ‘—β‰ˆπ‘—.
Translate doubling ensures that the balls π΅ξ…žξ…žπ‘—,𝑖 are of comparable measure with 𝐡𝑗≑𝐡𝑙(π‘œ,π‘Ÿπ‘—), so there exists a constant 𝐢>0 such that 𝑓(π‘Ÿπ‘—+1)β‰₯2𝑗𝑓(π‘Ÿπ‘—)/𝐢 for each π‘—βˆˆβ„•. Iterating this, we get that 𝑓(π‘Ÿπ‘—)β‰₯2𝑗(π‘—βˆ’1)/2𝑓(π‘Ÿ1)/𝐢𝑗. Since logπ‘Ÿπ‘—β‰ˆπ‘—, the result follows.

We omit the proof of Theorem 4.3 as it is so similar to that of Theorem 4.2. In fact it differs from it only in the last paragraph above, and the required modifications are straightforward.

We now consider some examples. All of our examples are either 𝑛-dimensional Riemannian manifolds or one-point joins of a finite number of 𝑛-dimensional Riemannian manifolds (meaning that the distinguished points π‘œ in these manifolds are all identified with each other). In all these cases, the associate measure is the usual measure on a Riemannian manifold (or equivalently Hausdorff 𝑛-measure).

It is easy to give examples relevant to Theorem 4.1. Euclidean space ℝ𝑛 has spherical boundary of cardinality 2 for 𝑛=1, and 1 for 𝑛>1: the 𝑛=1 case follows easily from the definition, while the 𝑛>1 case follows for instance from Corollary 3.4. The one-point join at 0 of π‘˜βˆˆβ„• copies of the half-line [0,∞) is a doubling space whose spherical boundary has cardinality π‘˜.

We do not know whether or not there exists a space (𝑋,𝑙,πœ‡) that satisfies the assumptions of Theorem 4.2 and has sharp volume growth rateliminfπ‘Ÿβ†’βˆžξ€·π΅logπœ‡π‘™ξ€Έ(π‘œ,π‘Ÿ)log2π‘Ÿ<∞.(4.4) However, hyperbolic space (𝐻𝑛,𝑙,πœ‡) is an example of a translate doubling space with much faster volume growth whose spherical boundary has infinite Minkowski dimension, as follows from the following more precise result.

Proposition 4.4. Let 𝜎=𝔖(𝑙,π‘œ,𝑔) be the sphericalized metric on 𝐻𝑛 for the standard sphericalizing function 𝑔(𝑑)=(1+𝑑)βˆ’2. The minimum number π‘π‘Ÿ of 𝜎-balls of radius π‘Ÿ>0 required to cover πœ•π‘†π»π‘› satisfies πΆπ‘›βˆ’1π‘Ÿπ‘›βˆ’1ξ‚΅exp(π‘›βˆ’1)π‘Ÿξ‚Άβ‰€π‘π‘Ÿβ‰€πΆπ‘›π‘Ÿπ‘›βˆ’1ξ‚΅exp2(π‘›βˆ’1)π‘Ÿξ‚Ά,(4.5) where 𝐢𝑛 depends only on 𝑛. For a general sphericalizing function 𝑔, π‘π‘Ÿ grows faster than 𝑐𝑔⋅exp((π‘›βˆ’1)/π‘Ÿπ‘π‘”)/𝐢𝑛, where 𝑐𝑔,𝐢𝑛>0 depend only on their subscripted parameters.

Proof. We assume that 𝑛=2: this does not change anything essential in the proof but it simplifies the notation. It suits us to think of 𝐻2 as the warped product 𝐡×𝑓𝐹, where 𝐡=[0,∞), 𝐹=𝑆1, and the warping function is 𝑓(𝑑)=sinh𝑑. We identify πœ•π‘†π»2 with 𝐹 as a set and view 𝐹 as the set of points in the complex plane of the form exp(π‘–πœƒ),πœƒβˆˆβ„. For the moment, assume that 𝑔(𝑑)=(1+𝑑)βˆ’2. Due to the symmetry of 𝐻2, to get the lower bound on π‘π‘Ÿ, it suffices to show that 𝜎(exp(π‘–πœƒ),1)β‰₯2π‘Ÿ, whenever πœƒβ‰ˆexp(βˆ’1/π‘Ÿ)/π‘Ÿ, is sufficiently small.
Suppose we join exp(π‘–πœƒ) with 1 via a path 𝛾 whose 𝐡-coordinate achieves a minimum value 𝑠β‰₯0. Considering only the horizontal component of arclength, we deduce from that len𝜎∫(𝛾)β‰₯2βˆžπ‘ π‘”(𝑑)𝑑𝑑=2/(1+𝑠). Considering only the vertical component of arclength, we have len𝜎(𝛾)β‰₯𝐻(𝑠)∢=πœƒπ‘”(𝑠)sinh𝑠. Thus 𝜎(exp(π‘–πœƒ),0)β‰₯π‘š, where π‘š is the minimum over all 𝑠β‰₯0 of 𝑀(𝑠)∢=2/(1+𝑠)∨𝐻(𝑠). Since we may takeπœƒ to be less than πœƒ0 for any πœƒ0>0 of our choice, we may assume that the minimum of 𝑀(𝑠) occurs when 𝑠>1. But then π‘š equals the minimum over all 𝑠β‰₯1 of 2(1+𝑠)βˆ’1∨(πœƒ(1+𝑠)βˆ’2sinh𝑠), which occurs when 1+𝑠=πœƒsinh(𝑠)/2. Taking π‘Ÿ=1/(1+𝑠) in this last equation gives πœƒ=2sinh(1βˆ’1/π‘Ÿ)/π‘Ÿ, as required.
To obtain an upper bound on π‘π‘Ÿ, it suffices to consider the path 𝛾 consisting of a horizontal segment from exp(π‘–πœƒ) to the point with first coordinate 𝑠 where 𝑠=πœƒsinh(𝑠), then the shorter vertical segment to the point with second coordinate 1, and finally a horizontal segment to 1∈𝐹=πœ•π‘†π»2. Then 𝜎(exp(π‘–πœƒ),1)≀len𝜎(𝛾)=2/(1+𝑠)+𝐻(𝑠), where 𝐻(𝑠) is as above. Taking 𝑠 as above gives the required upper bound for π‘π‘Ÿ.
For a general sphericalizing function 𝑔, we obtain as above that len𝜎(𝛾)≳𝐺(𝑠)∨(πœƒπ‘”(𝑠)sinh𝑠),(4.6) where 𝐺(𝑠)=(1+𝑠)𝑔(𝑠). This lower bound is minimal when 1+𝑠=πœƒsinh𝑠. Using the fact that 𝐺(𝑠) decays at a polynomial rate as π‘ β†’βˆž, it is a routine matter to obtain the desired conclusion.

Finally we show that Theorem 4.3 is sharp by considering the warped product (𝑋,𝑙), where 𝑋=𝐡×𝑓𝐹,𝐡=[0,∞),𝐹=𝑆1, the warping function is 𝑓(𝑑)=𝑑2, and the sphericalizing function is 𝑔(𝑑)=(1+𝑑)βˆ’2. We take π‘œβˆˆπ‘‹ to be the (unique) point with first coordinate 0.

Proposition 4.5. If (𝑋,𝑙,π‘œ) is as above then (πœ•π‘†π‘‹,𝜎) is bilipschitz equivalent to the arclength metric on 𝑆1. Moreover, (𝑋,𝑙,πœ‡) is weak translate doubling, where πœ‡ is Hausdorff 2-measure.

Proof. As in the proof of Proposition 4.4, a lower bound on 𝜎(exp(π‘–πœƒ),0) is given by the minimum π‘š over all 𝑠β‰₯0 of 2/(1+𝑠)∨𝐻(𝑠), where 𝐻(𝑠)=πœƒπ‘“(𝑠)𝑔(𝑠). Taking πœƒ>0 to be small, we may assume that the minimum occurs when 𝑠>1. But then π‘š is comparable with the minimum over all 𝑠β‰₯1 of π‘ βˆ’1βˆ¨πœƒ, which occurs when 𝑠=1/πœƒ and equals πœƒ. Thus 𝜎(exp(π‘–πœƒ),0)β‰³πœƒ.
On the other hand, as in Proposition 4.4, we see that 2𝜎(exp(π‘–πœƒ),1)≀11+1/πœƒ+πœƒπ‘”πœƒξ‚π‘“ξ‚€1πœƒξ‚β‰ˆπœƒ,0<πœƒβ‰€πœ‹.(4.7) The fact that 𝑋 is weak translate doubling follows from the fact that there exists constants 𝑐,𝐢>0 such that 𝑐(π‘Ÿ2βˆ§π‘Ÿ3)β‰€πœ‡(π΅π‘Ÿ)≀𝐢(π‘Ÿ2βˆ¨π‘Ÿ3) whenever π΅π‘Ÿ is a ball of radius π‘Ÿ>0. We leave this as an exercise to the reader.

Note that the space 𝑋 in Proposition 4.5 and Euclidean 3-space have the same volume growth rate, but πœ•π‘†π‘‹ is topologically 𝑆1 (at least for 𝑔 decaying no faster than the standard sphericalizing function) whereas πœ•π‘†β„3 is a one-point space. This again emphasizes that although the size of the spherical boundary constrains volume growth (for a large class of spaces), volume growth does not determine the size of the spherical boundary. We have also seen that restrictions such as negative Ricci curvature in the case of Riemannian manifolds is also not sufficient to ensure a nontrivial boundary. We would need more detailed curvature conditions, like an upper bound on the decay rate of Alexandrov curvature, in order to obtain results in that direction.

Acknowledgments

Both authors were partially supported by Enterprise Ireland, and the first author was partially supported by Science Foundation Ireland.

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