# The Spherical Boundary and Volume Growth

Authors: Stephen M. Buckley; Simon L. Kokkendorff
Journal: ISRN Geometry
 (2012)

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## Abstract

We consider thespherical 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 “small” (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 “norm-like” product metrics on a (possibly infinite) product of unbounded length spaces, is annular convex.


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## Body

## 1. Introduction

There are various notions of “boundaries at infinity’’ of metric spaces in the literature. One of these is the spherical boundary∂SX 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∂SX 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 ∂SX 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 Section2, 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 bya∧b and a∨b the minimum and maximum, respectively, of numbers a, 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 X¯d the metric closure of (X,d) and, viewing X as a subset of X¯d, we write ∂Xd=X¯d∖X. Given x,y∈X¯d,Γ(x,y) denotes the class of rectifiable paths λ:[0,T]→X¯d for which λ|(0,T) is a rectifiable path in X,λ(0)=x, and λ(T)=y. We also define Γd(x,y) to be the subset of Γ(x,y) consisting of paths that are parametrized by d-arclength. We write Γ(x,y;X) or Γd(x,y;X) if the space needs to be specified.Suppose(X,d) is rectifiably connected and, only for this paragraph, let us write d′(x,y)=infγ∈Γd(x,y)lend(γ), x,y∈X¯d. When restricted to X×X, d′ 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(x,y)=d′(x,y), for all x,y∈X; this equality clearly extends to points x,y∈X¯d. More generally, we say that a rectifiably connected metric space (X,d) is a local length space if d(x,y)=d′(x,y) whenever x∈X, y∈X¯d, and d(x,y)≤d(x,∂Xd). (X,d) is a geodesic space if, for all x,y∈X, there exists a path γ∈Γd(x,y) of length d(x,y).Every domainΩ⊂ℝn 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),∂0Xd, to be the set of all points y∈∂Xd for which Γd(x,y) is nonempty for some (and hence all) x∈X. Equivalently ∂0Xd is the set of all y∈X¯d whose inner distance from some (and hence all) x∈X is finite. If d is a length metric, then ∂0Xd=∂Xd, but equality may fail if d is merely a local length space. For instance, if d is the Euclidean metric on a domain Ω⊂ℝn which spirals sufficiently tightly near some point y∈∂Ωd, then y∉∂0Ωd.The rest of our notation is quite standard. We denote byBd(x,r), B¯d(x,r), and Sd(x,r), the open ball, closed ball, and sphere of radiusraboutx∈X; we omit the d-subscript if the metric is understood. If r≤0, Bd(x,r) is the empty set. A metric space is proper if all its closed balls are compact.Anarc in X is an injective path γ:I→X. We do not distinguish notationally between paths and their images. If γ is an arc in X, and u,v∈γ, γ[u,v] is the subarc of γ with endpoints u, v. Given two d-rectifiable arcs γ,γ′ in metric spaces (X,d) and (X′,d′) with lend(γ)≤lend′(γ′), we define the initial length mapf:γ→γ′ by the requirement that f maps each initial segment γ[u,v] of γ to the initial segment γ′[u′,v′] of γ′ that satisfies lend′(γ′[u′,v′])=lend(γ[u,v]).


### 2.2. Metric Measure Spaces

Ametric measure space(X,d,μ) is a metric space with an attached positive Borel measure μ which gives positive finite measure to all balls; if (X,d) is a (local) length space we call (X,d,μ) a (local) length measure space.Suppose(X,d,μ) is a metric measure space and C≥1. We say that (X,d,μ) is C-doubling if μ(2B)≤Cμ(B) whenever tB=B(x,tr) for fixed but arbitrary x∈X, r>0. We say that (X,d,μ) is C-translate doubling if instead μ(B′)≤Cμ(B) whenever B, B′ are overlapping balls of the same radius or weakC-translate doubling if we merely have μ(B′)≤C(1+r)μ(B) whenever B, B′ are overlapping balls of radius r.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 Hausdorffn-measure on hyperbolic n-space and arclength measure on the Cayley graph of an n-generator free-group.Thelower Minkowski dimensiondim̲ME of a subset E of a metric space (X,d) is defined by(2.1)dim⁡̲M(E)=liminf⁡ϵ→0+log⁡N(E,ϵ)log⁡(1/ϵ),
where N(E,ϵ) is the maximum cardinality of a collection of disjoint open balls of radius ϵ and centers in E.


### 2.3. The Spherical Boundary

A Borel functiong:[0,∞)→(0,∞) is said to be a C-sphericalizing function, C>2, if it has the following properties:(S1)
g(r)≤Cg(s) whenever r,s≥0,r≤2s+1,  and s≤2r+1;(S2)
∫r∞g(t)dt≤Crg(r),r≥1.We recall the following property of a sphericalizing function, taken from [1].Lemma 2.1.
Ifg:[0,∞)→(0,∞) is a C-sphericalizing function then(S3)
g(s)/g(r)≤C2(r/s)1+1/C,   for all 1≤r≤s.
In particular,tg(t)→0 as t→∞.Suppose(X,l,o) is an unbounded pointed local length space, and let us write |x|=l(x,o), x∈X¯l. Given a sphericalizing function g:[0,∞)→(0,∞), we define a new metric 𝔖(l,o,g) on X by the equation:(2.2)S(l,o,g)(x,y)=inf⁡γ∈Γ(x,y)∫γg(|z|)dl(z),x,y∈X.
We usually write σ in place of 𝔖(l,o,g).Ifγ∈Γl(x,y), x∈X, y∈∂Xl, then it is clear that γ is also of finite σ-length, so the length boundary ∂0Xl can be viewed as a subset of ∂Xσ. We define the spherical boundary ofX,∂SX to be ∂Xσ∖∂0Xl, and the spherical closure ofX to be X¯σ. Since any point in x∈X∪∂0Xl is at a finite l-distance from o, 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 X∪∂0Xl if and only if it is l-convergent. It also follows that if x∈X, y∈∂SX, and γ∈Γσ(x,y), then γ cannot be contained in any ball Bl(o,r).We record some useful elementary estimates involving the two metricsl and σ=𝔖(l,o,g). Below, G(t)≡(1+t)g(t), |x|≡l(x,o), δ∞(x)=σ(x,∂S(X)), and Cg is the sphericalization constant of g. It follows by standard analysis (as in [1, Proposition 2.14]) that δ∞(x)≥∫|x|∞g(t)dt, for all x∈X. Using (S2) and (S3), we readily deduce that(2.3)δ∞(x)≥Cg-2G(|x|),x∈X.
Suppose γ∈Γl(o,x) for some x∈X, with lenl(γ)=L<|x|+1, and let ν=γ|[s,L], where γ(s)=v,|v|≥1. Then |γ(t)|∈(t-1,t), 0≤t≤L, and so using (S1) and (S2), we deduce that(2.4)lenσ(ν)=∫sLg(|γ(t)|)dt≤Cg2∫|v|Lg(t)dt≤Cg3G(|v|).


## 2.1. Metric Spaces and Paths

Below,(X,d) is always a metric space which may have additional properties as specified. We denote by X¯d the metric closure of (X,d) and, viewing X as a subset of X¯d, we write ∂Xd=X¯d∖X. Given x,y∈X¯d,Γ(x,y) denotes the class of rectifiable paths λ:[0,T]→X¯d for which λ|(0,T) is a rectifiable path in X,λ(0)=x, and λ(T)=y. We also define Γd(x,y) to be the subset of Γ(x,y) consisting of paths that are parametrized by d-arclength. We write Γ(x,y;X) or Γd(x,y;X) if the space needs to be specified.Suppose(X,d) is rectifiably connected and, only for this paragraph, let us write d′(x,y)=infγ∈Γd(x,y)lend(γ), x,y∈X¯d. When restricted to X×X, d′ 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(x,y)=d′(x,y), for all x,y∈X; this equality clearly extends to points x,y∈X¯d. More generally, we say that a rectifiably connected metric space (X,d) is a local length space if d(x,y)=d′(x,y) whenever x∈X, y∈X¯d, and d(x,y)≤d(x,∂Xd). (X,d) is a geodesic space if, for all x,y∈X, there exists a path γ∈Γd(x,y) of length d(x,y).Every domainΩ⊂ℝn 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),∂0Xd, to be the set of all points y∈∂Xd for which Γd(x,y) is nonempty for some (and hence all) x∈X. Equivalently ∂0Xd is the set of all y∈X¯d whose inner distance from some (and hence all) x∈X is finite. If d is a length metric, then ∂0Xd=∂Xd, but equality may fail if d is merely a local length space. For instance, if d is the Euclidean metric on a domain Ω⊂ℝn which spirals sufficiently tightly near some point y∈∂Ωd, then y∉∂0Ωd.The rest of our notation is quite standard. We denote byBd(x,r), B¯d(x,r), and Sd(x,r), the open ball, closed ball, and sphere of radiusraboutx∈X; we omit the d-subscript if the metric is understood. If r≤0, Bd(x,r) is the empty set. A metric space is proper if all its closed balls are compact.Anarc in X is an injective path γ:I→X. We do not distinguish notationally between paths and their images. If γ is an arc in X, and u,v∈γ, γ[u,v] is the subarc of γ with endpoints u, v. Given two d-rectifiable arcs γ,γ′ in metric spaces (X,d) and (X′,d′) with lend(γ)≤lend′(γ′), we define the initial length mapf:γ→γ′ by the requirement that f maps each initial segment γ[u,v] of γ to the initial segment γ′[u′,v′] of γ′ that satisfies lend′(γ′[u′,v′])=lend(γ[u,v]).


## 2.2. Metric Measure Spaces

Ametric measure space(X,d,μ) is a metric space with an attached positive Borel measure μ which gives positive finite measure to all balls; if (X,d) is a (local) length space we call (X,d,μ) a (local) length measure space.Suppose(X,d,μ) is a metric measure space and C≥1. We say that (X,d,μ) is C-doubling if μ(2B)≤Cμ(B) whenever tB=B(x,tr) for fixed but arbitrary x∈X, r>0. We say that (X,d,μ) is C-translate doubling if instead μ(B′)≤Cμ(B) whenever B, B′ are overlapping balls of the same radius or weakC-translate doubling if we merely have μ(B′)≤C(1+r)μ(B) whenever B, B′ are overlapping balls of radius r.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 Hausdorffn-measure on hyperbolic n-space and arclength measure on the Cayley graph of an n-generator free-group.Thelower Minkowski dimensiondim̲ME of a subset E of a metric space (X,d) is defined by(2.1)dim⁡̲M(E)=liminf⁡ϵ→0+log⁡N(E,ϵ)log⁡(1/ϵ),
where N(E,ϵ) is the maximum cardinality of a collection of disjoint open balls of radius ϵ and centers in E.


## 2.3. The Spherical Boundary

A Borel functiong:[0,∞)→(0,∞) is said to be a C-sphericalizing function, C>2, if it has the following properties:(S1)
g(r)≤Cg(s) whenever r,s≥0,r≤2s+1,  and s≤2r+1;(S2)
∫r∞g(t)dt≤Crg(r),r≥1.We recall the following property of a sphericalizing function, taken from [1].Lemma 2.1.
Ifg:[0,∞)→(0,∞) is a C-sphericalizing function then(S3)
g(s)/g(r)≤C2(r/s)1+1/C,   for all 1≤r≤s.
In particular,tg(t)→0 as t→∞.Suppose(X,l,o) is an unbounded pointed local length space, and let us write |x|=l(x,o), x∈X¯l. Given a sphericalizing function g:[0,∞)→(0,∞), we define a new metric 𝔖(l,o,g) on X by the equation:(2.2)S(l,o,g)(x,y)=inf⁡γ∈Γ(x,y)∫γg(|z|)dl(z),x,y∈X.
We usually write σ in place of 𝔖(l,o,g).Ifγ∈Γl(x,y), x∈X, y∈∂Xl, then it is clear that γ is also of finite σ-length, so the length boundary ∂0Xl can be viewed as a subset of ∂Xσ. We define the spherical boundary ofX,∂SX to be ∂Xσ∖∂0Xl, and the spherical closure ofX to be X¯σ. Since any point in x∈X∪∂0Xl is at a finite l-distance from o, 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 X∪∂0Xl if and only if it is l-convergent. It also follows that if x∈X, y∈∂SX, and γ∈Γσ(x,y), then γ cannot be contained in any ball Bl(o,r).We record some useful elementary estimates involving the two metricsl and σ=𝔖(l,o,g). Below, G(t)≡(1+t)g(t), |x|≡l(x,o), δ∞(x)=σ(x,∂S(X)), and Cg is the sphericalization constant of g. It follows by standard analysis (as in [1, Proposition 2.14]) that δ∞(x)≥∫|x|∞g(t)dt, for all x∈X. Using (S2) and (S3), we readily deduce that(2.3)δ∞(x)≥Cg-2G(|x|),x∈X.
Suppose γ∈Γl(o,x) for some x∈X, with lenl(γ)=L<|x|+1, and let ν=γ|[s,L], where γ(s)=v,|v|≥1. Then |γ(t)|∈(t-1,t), 0≤t≤L, and so using (S1) and (S2), we deduce that(2.4)lenσ(ν)=∫sLg(|γ(t)|)dt≤Cg2∫|v|Lg(t)dt≤Cg3G(|v|).


## 3. Spherical Boundaries and Annular Convexity

A metric space(X,d) is said to be C-annular convex for some C≥2 if for every o∈X, r>0, and every pair of points x1,x2∈B(o,r)∖B(o,r/2) there exists a path γ from x1 to x2 in the annulus B(o,Cr)∖B(o,r/C) of length at most Cd(x1,x2).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 countablel2 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,o) is a singleton set.Proof.
SinceX is proper, it follows from [5, Theorem 2.4] that ∂SX is nonempty. We write |a|=l(a,o), a∈X, and let σ=𝔖(l,o,g), where g is a given Cg-sphericalizing function.
Fixing a fix a pair of pointsz,w∈∂SX, we pick sequences (zn) and (wn) in X converging to z and w, respectively. Since necessarily |zn|→∞ and |wn|→∞, we may assume that |zn|>n and |wn|>n, n∈ℕ.
Joiningzn to o by a path of length at most |zn|+1, and picking a point zn′ on this path with the property that |zn′|=n, it follows from (2.4) that σ(zn,zn′)≤Cg3ng(n). Using Lemma 2.1, we deduce that σ(zn,zn′)→0 as n→∞. We similarly find points wn′ such that |wn′|=n and σ(wn,wn′)→0 as n→∞.
But by annular convexity and the properties of sphericalizing functions, it is easy to see thatσ(zn′,wn′)≤C′ng(n), where C′=C′(C,Cg) and so σ(zn′,wn′)→0. Thus σ(zn,wn)→0 and so z=w, as required.Note that the assumption thatX 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 n∈ℕ, joined together by identifying with each other the left endpoints of all such intervals, then it is easy to show that ∂SX is empty.It is clear from the proof of Proposition3.1 that C-annular convexity can be replaced by the following formally weaker condition: a metric space (X,d) is weaklyC-annular convex, where C≥1, if for every o∈X, r>0, and every pair of points x1, x2∈X such that d(o,xi)=r,i=1,2, there exists a path γ from x1 to x2 in X∖B(o,r/C) of length at most 2Cr. 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 weakC-annular convex, then it is (3C)-annular convex.Proof.
Consider distinct pointsx1,x2∈B(o,r)∖B(o,r/2). Let di:=d(o,xi), i=1,2. Join x1,x2 by a path λ of length less than d(x1,x2)+ϵ, where ϵ>0 is so small that d(x1,x2)+ϵ<2d(x1,x2). Clearly λ remains inside B(o,3Cr), so it certainly verifies the (3C)-annular convexity condition if it remains outside B(o,r/3). Assume therefore that λ ventures inside this ball. Let z1, z2 be the first and last points z on λ such that d(z,o)=r/3, let γ1 be the initial segment of λ from x1 to z1, let γ2 be the final segment of λ from z2 to x2, and let γ3 be a path given by weak annular convexity for the pair z1, z2 and center point o. Let γ be the concatenation of γ1, γ3, and γ2.
Thenlend(γ1)+lend(γ2)≤ϵ+d(x1,x2) and weak annular convexity gives lend(γ3)≤2Cr/3, so lend(γ)≤ϵ+d(x1,x2)+2Cr/3. The path λ intersects B(o,r/3) and its endpoints lie outside B(o,r/2), so r/3=2(r/2-r/3)<d(x1,x2)+ϵ. Thus
(3.1)lend(γ)<(2C+1)(d(x1,x2)+ϵ)≤3Cd(x1,x2),
when ϵ>0 is sufficiently small. By construction, γ remains outside B(o,r/3C) and, since λ⊂B(o,3Cr), it suffices to verify that γ3 also lies in this ball. But len(γ3)≤2Cr/3, and the endpoints of γ3 are a distance r/3 from o, so
(3.2)γ3⊂B̅(o,(C+1)r3)⊂B(o,3Cr),
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 Propositions3.1 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 Proposition3.3, 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,∥·∥) which we assume to consist of functions defined on an index set I. We say that the norm ∥·∥ on V is monotonic onI if(a)
V is a space of real-valued functions on I, that is, V⊆ℝI.(b)
Iff∈V, g:I→ℝ, and |g(i)|≤|f(i)| for i∈I, then g∈V and ∥g∥≤∥f∥.Assume now that(Xi,di), i∈I are metric spaces and let P:=∏i∈IXi. We define as follows the metric subproduct(X,d) of (Xi,di), i∈I, relative to some fixed a∈P and the norm ∥·∥, which is monotonic on I: if x=(xi)i∈I∈P, then x∈X whenever f(i):=di(ai,xi) defines a function f∈V, and we define d(x,y)=∥i↦di(xi,yi)∥, which makes sense by monotonicity of the norm. We write (X,d)=∏i∈IV;a(Xi,di).Theorem 3.5.
SupposeV is a normed vector space of dimension at least 2 which is monotonic on I, and suppose (X,d)=∏i∈IV;a(Xi,di), where each (Xi,di) is an unbounded length space. Then (X,d) is 4-annular convex.Before proving Theorem3.5, we discuss metric subproducts X and give some examples. If x,x′∈X, it is clear that d(x,x′)≤d(x,a)+d(a,x′), so certainly d(x,x′)<∞. It is routine to verify that d is a metric on X. Although X is defined with respect to some a∈P, it is clear that we get the same metric space (X,d) if a is replaced by any b∈X. However, we get a metric subproduct disjoint from our original X if we replace a by any b∈P∖X.Lemma 3.6.
SupposeV 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 i∈I. Then (X,d)=∏i∈IV;a(Xi,di) is also a length space.Proof.
Supposex,x′∈X and let ϵ>0 be fixed but arbitrary. We define a path λ:[0,1]→X such that λ(t)=(λi(t))i∈I,0≤t≤1, with the following important properties which we record for later reference:
(3.3)λi:[0,1]⟶Xiisaconstantspeedpathfromxitoxi′,i∈I,lend(λi)≤(1+ϵ)di(xi,xi′),i∈I,(3.4)d(λ(s),λ(t))≤(1+ϵ)(t-s)d(x,x′),0≤s≤t≤1.
In fact this is quite easy to do: since Xi is a length space, we can certainly pick λi satisfying (3.3). Then
(3.5)di(λi(s),λi(t))≤lendi(λi|[s,t])=(t-s)lendi(λi)≤(1+ϵ)(t-s)di(xi′,yi′),
for all i∈I and 0≤s≤t≤1. Assembling together these paths λi to get a path λ:[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(λ)≤(1+ϵ)d(x,x′). Since x,x′∈X and ϵ>0 are all arbitrary, the result follows.Suppose we fixo∈X, where o=(oi)i∈I. For every i∈I and R>0 we can find a point zi(R)∈Xi such that di(oi,zi(R))=R: in fact the unboundedness of Xi ensures that there exists z∈Xi such that di(oi,z)≥R, and then we use continuity to pick the required zi(R) on a path from z to oi. It follows that if v∈V, we can find a point xv=(xiv)i∈I∈X such that di(xiv,oi)=|vi| and so d(xv,o)=∥v∥.We use function notationf(i) for V versus subscript notation xi for X to emphasize the difference between the normed space (V,∥·∥) and the subproduct (X,d).The simplest examples of monotonic normed spaces arelp spaces associated with finite or countably infinite I, for 1≤p≤∞. In the case of finite I, the subproduct X coincides as a set with the full Cartesian product ∏i∈IXi. In the special case where I has cardinality 2 and p=2, we deduce Proposition 3.3.Beyond the abovelp 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(·) type.Note that ifI=ℕ and each Xi is the real line, then the subproduct is merely the normed space V translated by a sequence a∈ℝℕ: thus these subproducts are all cosets of V, and any two such subproducts for different choices of a 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 whereXi is the metric subspace of the real line given by Xi={0}∪(⋃n∈ℕ{n,1/n}) for each i∈ℕ and V=l2. If a=(0)i∈ℕ, then it is readily verified that the metric subproduct X:=∏n∈ℕV;aXi has the cardinality of the continuum, whereas if a=(c)i∈ℕ for any fixed c∈Xi∖{0}, then X is a countable space.It can be shown that this dependence of the cardinality ofX on our choice of a does not occur when the spaces (Xi,di), i∈ℕ 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 Theorem3.5. 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 r=4,x=(0,0),y=(2,0), and o=(1+δ,1+δ) where 0<δ<1. Then every path from x to y intersects B(o,cr) for any c>1/4, as long as δ<4c-1.We now move on to the proof of Theorem3.5.Proof of Theorem3.5.
Letx,y∈B(o,r)∖B(o,r/2) be the pair of points for which we want to verify the 4-annular convexity condition (with other data r,o as usual), and let S:=d(x,o)+d(y,o), so that S>r. As in the proof of Proposition 3.2, a path γ connecting x and y of length at most (1+ϵ)d(x,y), where ϵ>0, verifies the 4-annular convexity condition for data (x,y,o,r) unless γ intersects B(o,r/4). We may therefore assume that this intersection occurs and so
(3.6)(1+ϵ)d(x,y)≥lend(γ)>(d(x,o)-r4)+(d(y,o)-r4).
Taking a limit as ϵ→0 we get d(x,y)≥S-r/2. In particular, d(x,y)>S/2.
Let us writeeA∈{0,1}I for the characteristic function of any A⊂I: thus eA(i)=1 if and only if i∈A. For x∈X and A⊂I, let xA∈X be defined by xiA=xi if i∈A, and xiA=oi if i∉A. For convenience, we write ei:=e{i} and xi:=x{i} for any i∈I.
SinceV has dimension at least 2, monotonicity readily implies that there are distinct indices j,k∈I such that the basic functions ej,ek lie in V.
We now define “scalar multiplication’’ onX, restricted to scalar values 0≤α≤1. Choosing λ:[0,1]→X to be as in the proof of Lemma 3.6, with (x,x′,ϵ)=(x,o,ϵ) and 0<ϵ≤1/3, we let α·x:=λ(s), where s is the minimal s′∈[0,1] such that d(λ(s′),o)=αd(x,o). This definition is typically not unique since λ is not unique, but note that if xi=oi then (α·x)i=oi for all 0≤α≤1.
Suppose first that we can find some such asA such that d(xA,o)≥r/4 and d(yB,o)≥r/4, where B:=I∖A. Let zx:=α·xA and zy:=β·yB for α=r/4d(xA,o) and β=r/4d(yB,o), so that d(zx,o)=d(zy,o)=r/4. Let λi, i∈A, and s be the associated coordinate paths and argument for α·xA, as in the last paragraph.
Now joinx and y by a path γ defined in the following piecewise manner by concatenating, in the natural order, paths γm:[0,1]→X, 1≤m≤4. First γ1 is a path from x to zx which has component paths γi1:[0,1]→Xi, where γi1 is a rescaled copy of λi|[0,s] if i∈A (and λ=(λi),s are as above), while γi1 is a constant speed path from xi to oi of length at most (1+ϵ)di(xi,oi) if i∈B. Thus, each γi is a constant speed path of length at most (1+ϵ)di(xi,oi), and so lend(γ1)≤(1+ϵ)d(x,o).
Next letzxy=(zixy)i∈I, zixy=zix for i∈A, and zi=ziy for i∈B. Then d(zx,zxy)=d(zy,zxy)=r/4. Let γ2:[0,1]→X be any path zx to z such that the coordinate paths γi2 are stationary paths for i∈A and are of di-length at most (1+ϵ)di(zix,zixy) for i∈B. Since ∥·∥ is a monotonic norm, we deduce that lend(γ2)≤(1+ϵ)r/4.
Finallyγ3 and γ4 are analogues of γ2 and γ1, respectively, but with (x,A,B) replaced by (y,B,A), and with the directions of the paths reversed. It follows that the d-length of our concatenated path γ is at most (1+ϵ)(S+r/2)≤3(1+ϵ)S/2. Since d(x,y)≥S/2 and ϵ≤1/3, we deduce that len(γ)≤4d(x,y), as required.
We next need to show thatγm, 1≤m≤4, stays outside B(o,r/4). It follows from the definition of α·x that d(λ1(t),o)≥r/4 for all 0≤t≤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 eachγm is contained in B(o,4r). The triangle inequality ensures that d(γ1(t),o)<r+(ϵ/2)r<4r and d(γ2(t),o)<(r/4)+(1+ϵ)(r/4)<4r for all 0≤t≤1. The same estimates for γ3 and γ4 follow by symmetry.
We may therefore make the added assumption that there is no way to splitI into complementary subsets A and B such that d(xA,o)≥r/4 and d(yB,o)≥r/4. Note though that for any w∈X and A⊂I, we either have d(wA,o)≥d(w,o)-r/4>r/4 or d(wI∖A,o)≥r/4. By our added assumption, it follows that for every A⊂I and B:=I∖A, either d(wA,o)≥d(w,o)-r/4 and d(wB,o)<r/4 both hold for w∈{x,y} or d(wB,o)≥d(w,o)-r/4 and d(wA,o)<r/4 both hold for w∈{x,y}.
In particular, one of these last pairs of conditions holds for a setA that contains j but not k. By switching the definitions of (A,j) and (B,k) if necessary, we assume that d(wA,o)≥d(w,o)-r/4 and d(wB,o)<r/4 both hold for w∈{x,y}, and that j∈A. We choose w∈X such that wi=oi for i≠k and d(w,o)=r/4.
As in the previous case, we letzx:=α·xA for α=r/4d(xA,o), but now we choose zy:=β·yA for β=r/4d(yA,o). As before d(zx,o)=d(zy,o)=r/4. Also let wx,wy be the points satisfying wix=zix and wiy=ziy if i≠k, and wkx=wky=wk.
We now joinx and y by a path γ defined in the following piecewise manner by concatenating, in the natural order, paths γm:[0,1]→X, 1≤m≤5, where γ1 is a path from x to zx defined as in the previous case, γ2 is a path of length at most (1+ϵ)r/4 from zx to wx which is stationary except in coordinate k, γ3 is a path of length at most (1+ϵ)r/2 from wx to wy which is stationary in coordinate k, γ4 is a path of length at most (1+ϵ)r/4 from wy to zy which is stationary except in coordinate k, and γ5 is analogous to γ1 in reverse, but from zy to y. As in the previous case, we see that len(γ)≤(1+ϵ)S+r. Since d(x,y)>S/2>r/2, by taking ϵ to be sufficiently small we get len(γ)≤4d(x,y), as required. The fact that γ⊂B(o,4r)∖B(o,r/4) can be verified as before, so we leave it to the reader.The examples that we have so far include the cases whereV 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(λi(t),oi) for different values of i∈I and fixed 0<t<1, where λ=(λi) 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 λi:[0,1]→Xi is a constant speed geodesic. Then λ=(λi) is also a constant speed geodesic and d(λ(t),o)=(1-t)d(x,o), 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 manifoldX=H2×H2 has exponential volume growth and constant negative Ricci curvature, but Corollary 3.4 implies that ∂SX is a one-point space.However, implications in the reverse direction are possible. OurGuiding Principle is that for reasonably general classes of pointed length measure spaces (X,l,o,μ), 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,o) is a pointed length space, g is a Cg-sphericalizing function, with associated spherical metric σ=𝔖(l,o,g) and spherical boundary ∂SX. Also G(t)=(1+t)g(t),|x|=l(x,o), and δ∞(x)=distσ(x,∂SX).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,μ) is Cμ-doubling. Then ∂SX is a finite set whose cardinality is bounded by a number dependent only on Cg and Cμ.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,μ) is Cμ-translate doubling. If dim̲M∂SX>0, then f(r)≡μ(Bl(o,r)) grows faster than any polynomial. In fact,
(4.1)liminf⁡r→∞log⁡f(r)log⁡2r>0.Theorem 4.3.
Suppose(X,l,μ) is weak Cμ-translate doubling. If dim̲M∂SX>0, then f(r):=μ(Bl(o,r)) grows at a polynomial rate or faster, that is,
(4.2)liminf⁡r→∞log⁡f(r)log⁡r>0.Proof of Theorem4.1.
Suppose∂SX has at least N points z1,…,zN. Choose 0<ϵ<(δ∞(o)/5)∧(G(1)/9Cg3) so small that the balls 4Bi≡Bσ(zi,4ϵ) are all disjoint and choose points ui∈Bi.
Supposeδ∞(z)≤3ϵ. Using (2.3), we get
(4.3)G(|z|)Cg2≤G(1)3Cg3.
It now follows from (S1) and the definition of G that |z|≥2.
We carry out the following construction for each indexi, 1≤i≤N. Choose γi∈Γl(o,ui) with lenl(γ)=L<|ui|+1, and let Bi′:=Bσ(vi,ϵ), where vi is the first point at which γi meets 3Bi¯. Now d∞(vi)≤3ϵ, and so |vi|≥2. Using (2.3) and (2.4), we see that G(|vi|)≈ϵ. In view of (S3), we see that the distances |vi| are mutually comparable, so let us choose a pair of mutually comparable radii r,R such that 2≤r≤|vi|≤R. By (S1), Bl(vi,t)⊂Bσ(vi,Cgtg(t)) for every 0<t≤r. We can therefore fix t≤r,t≈r,   so that Bi′′:=Bl(vi,t)⊂Bi′.
EveryBi′′ is contained in the single ball B0=Bl(o,R+t) and in turn B0 is contained in each of the balls sBi′′, s=(2R+t)/t. Since t and R are comparable, doubling ensures that μ(B0)≤C1μ(Bi′′), where C1 depends only on Cμ and (2R+t)/t≲1. Since B0 contains N disjoint balls of measure at least μ(B0)/C1, it follows that N≤C1, as required.Proof of Theorem4.2.
Part of the proof is similar to that of Theorem4.1, so we will be sketchy. Since dim̲M∂SX>0, there are constants c,Q>0 such that ∂SX contains cϵ-Q disjoint σ-balls Bϵ,i of radius ϵ for all 0<ϵ≤diaσ(X). Taking ϵj=A-jdiaσ(X) for a fixed number A>1, we can associate radii tj such that each of the cϵj-Q balls Bϵj,i contains an l-ball Bj,i′′ whose radius is tj and whose distance from the origin is contained in the interval [rj,Rj] for some numbers rj, Rj are comparable with tj; the constants of comparability can be taken to depend only Cg. We assume, as we may, that A is chosen so large that cϵj-Q≥2j, r1≥1, and Rj+2tj<rj+1, j∈ℕ. Note also that the ratios rj+1/rj are uniformly bounded by a constant dependent only on A and Cg, so that logrj≈j.
Translate doubling ensures that the ballsBj,i′′ are of comparable measure with Bj≡Bl(o,rj), so there exists a constant C>0 such that f(rj+1)≥2jf(rj)/C for each j∈ℕ. Iterating this, we get that f(rj)≥2j(j-1)/2f(r1)/Cj. Since logrj≈j, the result follows.We omit the proof of Theorem4.3 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 eithern-dimensional Riemannian manifolds or one-point joins of a finite number of n-dimensional Riemannian manifolds (meaning that the distinguished points o 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 Theorem4.1. 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 k∈ℕ copies of the half-line [0,∞) is a doubling space whose spherical boundary has cardinality k.We do not know whether or not there exists a space(X,l,μ) that satisfies the assumptions of Theorem 4.2 and has sharp volume growth rate(4.4)liminf⁡r→∞log⁡μ(Bl(o,r))log⁡2r<∞.
However, hyperbolic space (Hn,l,μ) 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.
Letσ=𝔖(l,o,g) be the sphericalized metric on Hn for the standard sphericalizing function g(t)=(1+t)-2. The minimum number Nr of σ-balls of radius r>0 required to cover ∂SHn satisfies
(4.5)Cn-1rn-1exp⁡((n-1)r)≤Nr≤Cnrn-1exp⁡(2(n-1)r),
where Cn depends only on n. For a general sphericalizing function g, Nr grows faster than cg·exp((n-1)/rcg)/Cn, where cg,Cn>0 depend only on their subscripted parameters.Proof.
We assume thatn=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 B×fF, where B=[0,∞), F=S1, and the warping function is f(t)=sinht. We identify ∂SH2 with F as a set and view F as the set of points in the complex plane of the form exp(iθ),θ∈ℝ. For the moment, assume that g(t)=(1+t)-2. Due to the symmetry of H2, to get the lower bound on Nr, it suffices to show that σ(exp(iθ),1)≥2r, whenever θ≈exp(-1/r)/r, is sufficiently small.
Suppose we joinexp(iθ) with 1 via a path γ whose B-coordinate achieves a minimum value s≥0. Considering only the horizontal component of arclength, we deduce from that lenσ(γ)≥2∫s∞g(t)dt=2/(1+s). Considering only the vertical component of arclength, we have lenσ(γ)≥H(s):=θg(s)sinhs. Thus σ(exp(iθ),0)≥m, where m is the minimum over all s≥0 of M(s):=2/(1+s)∨H(s). Since we may takeθ to be less than θ0 for any θ0>0 of our choice, we may assume that the minimum of M(s) occurs when s>1. But then m equals the minimum over all s≥1 of 2(1+s)-1∨(θ(1+s)-2sinhs), which occurs when 1+s=θsinh(s)/2. Taking r=1/(1+s) in this last equation gives θ=2sinh(1-1/r)/r, as required.
To obtain an upper bound onNr, it suffices to consider the path γ consisting of a horizontal segment from exp(iθ) to the point with first coordinate s where s=θsinh(s), then the shorter vertical segment to the point with second coordinate 1, and finally a horizontal segment to 1∈F=∂SH2. Then σ(exp(iθ),1)≤lenσ(γ)=2/(1+s)+H(s), where H(s) is as above. Taking s as above gives the required upper bound for Nr.
For a general sphericalizing functiong, we obtain as above that
(4.6)lenσ(γ)≳G(s)∨(θg(s)sinh⁡s),
where G(s)=(1+s)g(s). This lower bound is minimal when 1+s=θsinhs. Using the fact that G(s) decays at a polynomial rate as s→∞, it is a routine matter to obtain the desired conclusion.Finally we show that Theorem4.3 is sharp by considering the warped product (X,l), where X=B×fF,B=[0,∞),F=S1, the warping function is f(t)=t2, and the sphericalizing function is g(t)=(1+t)-2. We take o∈X to be the (unique) point with first coordinate 0.Proposition 4.5.
If(X,l,o) is as above then (∂SX,σ) is bilipschitz equivalent to the arclength metric on S1. Moreover, (X,l,μ) is weak translate doubling, where μ is Hausdorff 2-measure.Proof.
As in the proof of Proposition4.4, a lower bound on σ(exp(iθ),0) is given by the minimum m over all s≥0 of 2/(1+s)∨H(s), where H(s)=θf(s)g(s). Taking θ>0 to be small, we may assume that the minimum occurs when s>1. But then m is comparable with the minimum over all s≥1 of s-1∨θ, which occurs when s=1/θ and equals θ. Thus σ(exp(iθ),0)≳θ.
On the other hand, as in Proposition4.4, we see that
(4.7)σ(exp⁡(iθ),1)≤21+1/θ+θg(1θ)f(1θ)≈θ,0<θ≤π.
The fact that X is weak translate doubling follows from the fact that there exists constants c,C>0 such that c(r2∧r3)≤μ(Br)≤C(r2∨r3) whenever Br is a ball of radius r>0. We leave this as an exercise to the reader.Note that the spaceX in Proposition 4.5 and Euclidean 3-space have the same volume growth rate, but ∂SX is topologically S1 (at least for g decaying no faster than the standard sphericalizing function) whereas ∂Sℝ3 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.



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*Source: 484312-2012-03-26.xml*