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arXiv:0809.0746v1 [math.MG] 4 Sep 2008

DISTANCE GEOMETRY IN QUASIHYPERMETRIC

SPACES. III

PETER NICKOLAS AND REINHARD WOLF

Abstract. Let (X,d) be a compact metric space and let M(X)

denote the space of all finite signed Borel measures on X. Define

I: M(X) → R by

I(µ) =

XX

??

d(x,y)dµ(x)dµ(y),

and set M(X) = supI(µ), where µ ranges over the collection of

signed measures in M(X) of total mass 1. This paper, with two

earlier papers [Peter Nickolas and Reinhard Wolf, Distance geome-

try in quasihypermetric spaces. I and II], investigates the geomet-

ric constant M(X) and its relationship to the metric properties

of X and the functional-analytic properties of a certain subspace of

M(X) when equipped with a natural semi-inner product. Specif-

ically, this paper explores links between the properties of M(X)

and metric embeddings of X, and the properties of M(X) when X

is a finite metric space.

1. Introduction

Let (X,d) be a compact metric space and let M(X) denote the space

of all finite signed Borel measures on X. Let I: M(X) → R be defined

by

?

XX

and set

M(X) = supI(µ),

where µ ranges over M1(X), the collection of signed measures in M(X)

of total mass 1.

Our interest in this paper and its predecessors [10] and [11] is in the

properties of the geometric constant M(X). In [10], we observed that

if (X,d) does not have the quasihypermetric property, then M(X) is

I(µ) =

?

d(x,y)dµ(x)dµ(y),

2000 Mathematics Subject Classification. Primary 51K05; secondary 54E45,

31C45.

Key words and phrases. Compact metric space, finite metric space, quasihy-

permetric space, metric embedding, signed measure, signed measure of mass zero,

spaces of measures, distance geometry, geometric constant.

1

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2 PETER NICKOLAS AND REINHARD WOLF

infinite, and thus the context of our study for the most part is that of

quasihypermetric spaces. Recall (see [10]) that (X,d) is quasihyperme-

tric if for all n ∈ N, all α1,...,αn∈ R satisfying?n

x1,...,xn∈ X, we have

n

?

i,j=1

i=1αi= 0, and all

αiαjd(xi,xj) ≤ 0.

In the presence of the quasihypermetric property, a natural semi-

inner product space structure becomes available on M0(X), the sub-

space of M(X) consisting of all signed measures of mass 0. Specifically,

for µ,ν ∈ M0(X), we define

(µ | ν) = −I(µ,ν),

and denote the resulting semi-inner product space by E0(X). The

associated seminorm ? · ? on E0(X) is then given by

?µ? =?−I(µ)?1

The semi-inner product space E0(X) is in many ways the key to our

analysis of the constant M(X). In [10], we developed the properties of

E0(X) in a detailed way, exploring in particular the properties of sev-

eral operators and functionals associated with E0(X), some questions

related to its topology, and the question of completeness. Questions di-

rectly relating to the constant M(X) were only examined in [10] when

they had a direct bearing on this general analysis. In [11], we dis-

cussed maximal measures (measures which attain the supremum defin-

ing M(X)), sequences of measures which approximate the supremum

when no maximal measure exists and conditions implying or equivalent

to the finiteness of M(X).

In this paper, building on the above work, we discuss

(1) metric embeddings of X, both of a explicitly geometric type

and of a more abstract functional-analytic type, and

(2) the properties of M(X) when X is a finite metric space.

We assume here that the reader has read [10] and [11], and we repeat

their definitions and results here only as necessary.

2.

2. Definitions and Notation

Let (X,d) (abbreviated when possible to X) be a compact metric

space. The diameter of X is denoted by D(X). We denote by C(X)

the Banach space of all real-valued continuous functions on X equipped

with the usual sup-norm. Further,

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DISTANCE GEOMETRY IN QUASIHYPERMETRIC SPACES. III3

• M(X) denotes the space of all finite signed Borel measures

on X,

• M0(X) denotes the subspace of M(X) consisting of all mea-

sures of total mass 0,

• M1(X) denotes the affine subspace of M(X) consisting of all

measures of total mass 1,

• M+(X) denotes the set of all positive measures in M(X), and

• M+

set of all probability measures on X.

For x ∈ X, the atomic measure at x is denoted by δx.

The following two functionals on measures play a central role in our

work. If (X,d) is a compact metric space, then for µ,ν ∈ M(X), we

set

?

XX

and then

I(µ) = I(µ,µ).

1(X) denotes the intersection of M+(X) and M1(X), the

I(µ,ν) =

?

d(x,y)dµ(x)dν(y),

Also, a linear functional J(µ) on M(X) is defined for each µ ∈ M(X)

by J(µ)(ν) = I(µ,ν) for all ν ∈ M(X). For µ ∈ M(X), the function

dµ∈ C(X) is defined by

?

X

for x ∈ X.

For the compact metric space (X,d), we define

dµ(x) =d(x,y)dµ(y)

M(X) = sup?I(µ) : µ ∈ M1(X)?.

3. Metric Embeddings of Finite Spaces

Metric embeddings of various types have played a significant role in

work on the geometric properties of metric spaces. In section 3 of [10],

for example, we discussed briefly some connections between the quasi-

hypermetric property and L1-embeddability and between the quasihy-

permetric property and the metric embedding ideas of Schoenberg [13].

Also, embedding arguments based around and extending Schoenberg’s

ideas were used in [1] by Alexander and Stolarsky to obtain information

on M(X) when X is a subset of euclidean space, and in [3] by Assouad

to characterize the hypermetric property in finite metric spaces (see

section 5 below for the definition of the hypermetric property).

In this and the following section, we apply metric embedding argu-

ments to the analysis of the constant M. In this section, our arguments

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4 PETER NICKOLAS AND REINHARD WOLF

are for finite spaces, and are of a more or less explicitly geometric char-

acter, while in the following section, we use embedding arguments of

a functional-analytic character, and the results are for the case of a

general (usually compact) metric space.

As mentioned in section 3 of [10], Schoenberg [13] proved that a sep-

arable metric space (X,d) is quasihypermetric if and only if the metric

space (X,d

In particular, if X is a finite space, then (X,d) is quasihypermetric if

and only if (X,d

of suitable dimension. We will refer to an embedding of (X,d

euclidean space or in Hilbert space as a Schoenberg-embedding or, for

short, an S-embedding of X.

Our results in this section relate the metric properties of a space X

which are our main interest to the geometric properties of the S-

embeddings of X and to the existence of invariant measures on X

(see section 3 of [11]) of total mass 1.

First we have the following result, for the proof of which we make

use of some ideas developed by Assouad [3].

1

2) can be embedded isometrically in the Hilbert space ℓ2.

1

2) can be embedded isometrically in a euclidean space

1

2) in a

Theorem 3.1. Let (X,d) be a finite metric space. If M(X) < ∞, then

every S-embedding of X in a euclidean space lies on (the surface of ) a

sphere.

Proof. Suppose that X = {x1,...,xn} and that the S-embedding of X

into the euclidean space E = Rmmaps xito yi∈ E for i = 1,...,n.

We are seeking z ∈ E such that ?yi− z?2= ?yj− z?2for all i,j,

and it is easy to see that this relation holds for z ∈ E if and only if

?y1?2− ?yi?2= 2(y1− yi | z) for all i. Further, if we let T denote

the hyperplane {(t1,...,tn) :?n

relation holds if and only if

i=1ti= 0} in Rn, we see that the last

n

?

i=1

ti?yi?2= 2

? n

?

i=1

tiyi

????z

?

for all (ti) ∈ T. Let {ek: k = 1,...,m} be an orthonormal basis for E,

and define functionals u and vkfor k = 1,...,m on the hyperplane T

by setting

u(t) =

n

?

i=1

ti?yi?2

andvk(t) =

? n

?

i=1

tiyi

????ek

?

for t ≡ (ti) ∈ T. Then it is clear that there exists z ∈ E for which

the condition above holds if and only if there exist scalars {αk: k =

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DISTANCE GEOMETRY IN QUASIHYPERMETRIC SPACES. III5

1,...,m} such that

u(t) =

m

?

k=1

αkvk(t)

for all t ∈ T. We claim that this holds if and only if

m

?

k=1

kervk⊆ keru.

To see this, suppose first that?kervk ⊆ keru. Now there exist

r ≡ (ri) and sk≡ (s(i)

u(t) = (r | t)

for all t ∈ T. Hence?kervkis the orthogonal complement within T

of the subspace of T generated by the {sk}, and it follows that r lies

in the subspace generated by the {sk}. Thus u(t) =?m

all t ∈ T, for suitable scalars {αk}. The converse is clear, and so the

claim holds.

Suppose that s ≡ (si) satisfies?si= 1. Straightforward manipula-

tions then show that

u(s) =1

2

i,j=1

k) in T, for k = 1,...,m, such that

andvk(t) = (sk| t)

k=1αkvk(t) for

n

?

sisj?yi− yj?2+

n

?

i,j=1

sisj(yi| yj)

and

m

?

k=1

vk(s)2=

n

?

i,j=1

sisj(yi| yj),

giving

u(s) =1

2

n

?

i,j=1

sisj?yi− yj?2+

m

?

k=1

vk(s)2.

(Note that s is not in the domain T of the functionals u and vk as

defined earlier, but we use the same symbols to denote the functions

whose values on s are defined by the same expressions.)

Given t ≡ (ti) ∈ T, define s ≡ (si) ∈ Rnby setting s1= t1+ 1 and

si= tifor i = 2,...,n, so that?si= 1. Then we clearly have

u(t) = u(s) − ?y1?2

for each k. Hence if t ∈?kervk, we have

u(t) = u(s) − ?y1?2

1

2

i,j=1

andvk(t) = vk(s) − (y1| ek)

=

n

?

sisj?yi− yj?2+

m

?

k=1

vk(s)2− ?y1?2

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6PETER NICKOLAS AND REINHARD WOLF

=

1

2

n

?

i,j=1

n

?

i,j=1

n

?

i,j=1

sisjd(xi,xj) +

m

?

k=1

m

?

k=1

?vk(t) + (y1| ek)?2− ?y1?2

=

1

2

sisjd(xi,xj) +(y1| ek)2− ?y1?2

=

1

2

sisjd(xi,xj)

≤

1

2M(X).

But since this holds for all t ∈?kervkand M(X) is finite, we must

have u(t) = 0 for all t ∈?kervk. Thus?kervk⊆ keru, and the result

follows.

?

We show later (Theorem 4.9) that the above implication holds when

X is a general compact metric space, with the corresponding sphere

then lying in general in the Hilbert space ℓ2.

In [1], Alexander and Stolarsky made use of S-embeddings on spheres

to derive interesting results on M and related matters for subsets of eu-

clidean spaces. In the following result, we gather together some of their

observations, specialized to the case of finite spaces, but generalized to

the non-euclidean case, along with some new observations.

Recall (see [10]) that for a compact metric space (X,d), we write

M+(X) = sup{I(µ) : µ ∈ M+

Theorem 3.2. Let (X = {x1,...,xn},d) be a finite metric space, and

suppose that X is S-embedded as the set Y = {y1,...,yn} on a sphere S

of radius r in some euclidean space, where the S-embedding maps xi

to yifor i = 1,...,n. Then we have the following.

1(X)}.

(1) M(X) ≤ 2r2.

(2) There exists a maximal measure on X.

If further the S-embedding of X is into a euclidean space of minimal

dimension, then we have the following.

(3) M(X) = 2r2.

(4) M+(X) = 2(r2− s2), where s is the distance from the centre

of S to the convex hull of Y .

(5) If w1,...,wn∈ R are such that?n

a maximal measure on X if and only if?n

of S.

i=1wi= 1, then?n

i=1wiδxiis

i=1wiyiis the centre

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DISTANCE GEOMETRY IN QUASIHYPERMETRIC SPACES. III7

Proof. Suppose without loss of generality that the centre of the sphere S

is 0. If w1,...,wn∈ R satisfy?n

culation (cf. Lemma 3.2 of [1]) gives

n

?

i,j=1

i,j=1

and it follows that

i=1wi= 1, then a straightforward cal-

wiwjd(xi,xj) =

n

?

wiwj?yi− yj?2= 2r2− 2

????

n

?

i=1

wiyi

????

2

,

M(X) = 2r2− 2inf

????

n

?

i=1

wiyi

???

2

:

n

?

i=1

wi= 1

?

and that

M+(X) = 2r2− 2inf

????

n

?

i=1

wiyi

???

2

: w1,...,wn≥ 0 and

n

?

i=1

wi= 1

?

.

This gives (1), and then (2) follows by Theorem 4.11 of [11]. Now

assume that the S-embedding of X is into Rk, where k is the minimum

dimension possible, so that the affine hull of Y is Rk. Then there exist

w1,...,wnwith?n

from the expression derived above for M(X) that M(X) = 2r2, and

we have (3). The expression for M(X) also clearly gives (5). Finally,

since the distance s from the centre of S to the convex hull of Y is

n

?

i=1

the expression derived above for M+(X) gives (4).

i=1wi= 1 such that?n

i=1wiyi= 0, and it follows

inf

????

wiyi

??? : w1,...,wn≥ 0 and

n

?

i=1

wi= 1

?

,

?

Corollary 3.3. In the circumstances of the theorem,

(1) there is a unique maximal measure on X if and only if the S-

embedded set Y is affinely independent, and

(2) if the S-embedding is into a space of minimal dimension, then

the maximal measure on X given by the theorem is a probability

measure if and only if the centre of the sphere S is in the convex

hull of Y .

Proof. Suppose that the S-embedding is into a space of minimal di-

mension. Then by part (5) of the theorem, there is a unique maximal

measure on X if and only if 0 can be written as an affine combination

of y1...,ynin a unique way, and this is the case if and only if Y is a

maximal affinely independent set. By the argument used for part (3)

of the theorem, this is equivalent in the general case to the affine inde-

pendence of Y , giving (1). Assertion (2) is immediate from part (5) of

the theorem.

?

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8PETER NICKOLAS AND REINHARD WOLF

Now we can prove the result alluded to earlier which expresses met-

ric properties of X as equivalent geometric conditions on S-embeddings

of X and also as equivalent conditions on d-invariant measures of

mass 1 on X. (We prefer to speak of invariant measures of mass 1

here rather than of maximal measures—see section 3 of [11] for the rel-

evant definitions—but recall that by Theorem 3.1 of [11] these classes

of measures coincide in any compact quasihypermetric space.)

Theorem 3.4. Let (X,d) be a finite quasihypermetric space.

(1) The following conditions are equivalent.

(a) M(X) < ∞.

(b) There exists a d-invariant measure in M1(X).

(c) Some S-embedding of X in a euclidean space lies on a

sphere.

(d) Every S-embedding of X in a euclidean space lies on a

sphere.

(2) The following conditions are equivalent.

(a) M+(X) = M(X).

(b) There exists a d-invariant measure in M+

(c) Some S-embedding of X in a euclidean space of minimal

dimension lies on a sphere whose centre is in the convex

hull of the S-embedded set.

(d) Every S-embedding of X in a euclidean space of minimal

dimension lies on a sphere whose centre is in the convex

hull of the S-embedded set.

(3) The following conditions are equivalent.

(a) X is strictly quasihypermetric.

(b) There exists a unique d-invariant measure in M1(X).

(c) Some S-embedding of X in a euclidean space is an affinely

independent set.

(d) Every S-embedding of X in a euclidean space is an affinely

independent set.

1(X).

Proof. (1) Theorem 3.1 of [11] shows that (b) implies (a), Theorem 3.1

(of the present paper) shows that (a) implies (d), the result of Schoen-

berg [13] quoted before Theorem 3.1 shows that there exists an S-

embedding of X into a euclidean space, from which it follows that (d)

implies (c), and Theorem 3.2, with Theorem 3.1 of [11], shows that (c)

implies (b).

(2) Assume (a), and consider any S-embedding of X on a sphere

in a euclidean space of minimal dimension. Then using (a) and parts

(3) and (4) of Theorem 3.2, we find that the distance of the centre of

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DISTANCE GEOMETRY IN QUASIHYPERMETRIC SPACES. III9

the sphere from the convex hull of the embedded set is 0, and com-

pactness yields (d). That (d) implies (c) is shown as in part (1), and

Corollary 3.3, with Theorem 3.1 of [11], shows that (c) implies (b).

Assume (b) and let µ ∈ M+

of [11] shows that µ has value M(X). But since µ ∈ M+

that M+(X) = M(X), and we have (a).

(3) Corollary 3.3 and Theorem 3.1 of [11] show that (c) implies (b),

those results together with part (1) show that (b) implies (d), and the

fact that (d) implies (c) is shown as earlier. Assume that X is strictly

quasihypermetric. Then by Theorem 5.3 of [11] (see Theorem 5.1 be-

low), we have M(X) < ∞, so by part (1), there exists a d-invariant

µ ∈ M1(X), which is unique by part (4) of Theorem 3.1 of [11]. Thus,

(a) implies (b). Now assume that X is not strictly quasihypermetric. If

M(X) = ∞, then by Theorem 3.1 of [11], X has no d-invariant measure

of mass 1. If M(X) < ∞, then by part (1), there exists a d-invariant

µ ∈ M1(X). Since X is not strictly quasihypermetric, it follows from

parts (2) and (5) of Lemma 5.1 of [10] that there exists a non-zero

d-invariant measure ν ∈ M0(X) (which, by Theorem 5.3 of [11], has

value 0). It follows that µ+ν ∈ M1(X), that µ+ν is d-invariant, and

that µ + ν ?= µ, so that there is more than one d-invariant measure of

mass 1 on X. Thus, (b) implies (a), completing the proof.

1(X) be d-invariant. Then Theorem 3.1

1(X), it follows

?

Remark 3.5. In [3], Assouad develops characterizations of the hyper-

metric property and the property of L1-embeddability of a finite metric

space. A space has one of these properties if it can be S-embedded on

a sphere in euclidean space in such a way as to satisfy an additional

lattice-theoretical constraint, stronger in the second case than the first,

since L1-embeddability implies the hypermetric property (cf. our The-

orem 4.3 below, the proof of which can be adapted routinely to show

this). It follows by Theorem 3.2 that such spaces have M finite (cf. The-

orem 4.4). It would be interesting to know if there are characterizations

of these two properties in terms of invariant measures.

4. Metric Embeddings of General Spaces

We begin by noting the following result, which relates the value

of M on a general compact metric space X and the value of M on

the finite subsets of X (see also Theorem 4.7 below). The result can

easily be proved either by an argument similar to that needed to show

that (1) implies (3) in Theorem 3.2 of [10], or by adapting the proof of

Lemma 3.3 of [1].

Theorem 4.1. If X is a compact metric space, then M(X) is the

supremum of the values M(F) for finite subsets F of X.

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10PETER NICKOLAS AND REINHARD WOLF

Definition 4.2. Let (X,d) be a metric space. We say that X ad-

mits an L1-embedding if there exists a probability space (Ω,A,P) and

a mapping i: X → L1(Ω) such that d(x,y) = ?i(x) − i(y)? for all

x,y ∈ X. Further, we say that this embedding is uniformly bounded if

sup|i(x)(ω)| < ∞, where x and ω range over X and Ω, respectively.

Assertion (1) of the following result is well known, as is the stronger

assertion that an L1-embeddable space is hypermetric (for the defini-

tion of the hypermetric property, see section 5 below).

Theorem 4.3. Let X be a metric space admitting an L1-embedding.

Then we have the following.

(1) X is quasihypermetric.

(2) If additionally X is compact and the given L1-embedding is uni-

formly bounded, with |i(x)(ω)| ≤ K for some K ≥ 0 and for all

x ∈ X and ω ∈ Ω, then M(X) ≤ K.

Proof. (1) Consider n ∈ N, x1,...,xn∈ X and α1,...,αn∈ R such

that α1+ ··· + αn= 0. Then, since R is quasihypermetric, we have

n

?

i,j=1

i,j=1

n

?

i,j=1

≤ 0.

Therefore, X is quasihypermetric.

(2) Consider n ∈ N, x1,...,xn∈ X and α1,...,αn∈ R such that

α1+ ··· + αn= 1. As before, we have

n

?

i,j=1

i,j=1

Let K = sup{|i(x)(ω)| : x ∈ X,ω ∈ Ω}. By assumption, K < ∞. Ap-

plying Corollary 3.2 of [11] to the interval [−K,K] gives?n

i(xj)(ω)| ≤ K for all ω ∈ Ω, and so?n

fore, we have M(F) ≤ K for all finite subsets F of X, and it follows

by Theorem 4.1 that M(X) ≤ K.

We say that a real normed linear space (E,?·?) is quasihypermetric

if the corresponding metric space (E,d) is quasihypermetric, where d

is the norm-induced metric on E.

We wish next to discuss some properties of subsets of finite-dimensional

real normed linear spaces. Of course, every such space is isometrically

αiαjd(xi,xj) =

n

?

αiαj

??i(xi) − i(xj)??

??i(xi)(ω) − i(xj)(ω)???

=

?

Ω

?

αiαj

dPω

αiαjd(xi,xj) =

?

Ω

?

n

?

αiαj

??i(xi)(ω) − i(xj)(ω)???

dPω.

i,j=1αiαj|i(xi)(ω)−

i,j=1αiαjd(xi,xj) ≤ K. There-

?

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DISTANCE GEOMETRY IN QUASIHYPERMETRIC SPACES. III11

isomorphic to a space (Rn,?·?) for some n and some norm ?·?, and so it

suffices to restrict attention to subsets of spaces of this type. We recall

the well known fact that for any fixed space (Rn,? · ?), the following

three conditions are equivalent.

(1) The space (Rn,? · ?) is quasihypermetric.

(2) The space (Rn,? · ?) is isometrically isomorphic to a subspace

of L1([0,1]) (the space is L1-embeddable).

(3) The norm ? · ? admits a so-called L´ evy representation; that is,

there exist α > 0 and a probability measure P on the euclidean

unit sphere Sn−1in Rnsuch that

?

Sn−1

for all x ∈ Rn.

(For a proof, one can combine Corollaries 1.1 and 1.3 of [14] with

Corollaries 2.6 and 6.2 of [6].)

We have seen that M(X) may be infinite when X is a compact

(or even finite) quasihypermetric space. In the presence of a linear

structure, however, we have the following result, which generalizes the

euclidean case proved in Theorem 3.8 of [1].

?x? = α

??(x | ω)??dP(ω)

Theorem 4.4. Suppose that (Rn,? · ?) is quasihypermetric, and let

X be a subset of Rnwhich is compact when equipped with the norm-

induced metric. Then

(1) M(X) < ∞ and

(2) there exists c > 0 such that |I(µ1)−I(µ2)| ≤ c?µ1−µ2? for all

µ1,µ2∈ M+

Proof. Using the comments above, choose α > 0 and a probability

measure P on the euclidean unit sphere Sn−1in Rnsuch that

?

Sn−1

for all x ∈ Rn. Define i: X → L1(Sn−1,P) by setting i(x)(ω) = α(x | ω)

for x ∈ X and ω ∈ Sn−1.

x,y ∈ X, and, by the compactness of X, there exists K such that

|i(x)(ω)| = α|(x | ω)| ≤ α?x?2 ≤ K for all x ∈ X and ω ∈ Sn−1,

where ? · ?2 denotes the euclidean norm on Rn. Thus X admits a

uniformly bounded L1-embedding, and now Theorem 4.3 above and

part (6) of Theorem 5.3 of [10] (see also Remark 5.7 of [10]) complete

the proof.

1(X).

?x? = α

??(x | ω)??dP(ω)

Then ?x − y? = ?i(x) − i(y)? for all

?

Remark 4.5. It is well known that any finite metric space can be

isometrically embedded in Rnwith the ∞-norm for suitable n (see,

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12PETER NICKOLAS AND REINHARD WOLF

for example, part (1) of Lemma 2.2 of [10]), and it is also well known

that this normed space is non-quasihypermetric if n ≥ 3 (see section 3

of [10]). Thus it is the quasihypermetric property of the enclosing

normed space rather than of the embedded metric space that is crucial

for the conclusions of the theorem.

Theorem 4.6. Let (X,d) be a compact metric space with M(X) < ∞.

Then there exists a mapping i of X into separable Hilbert space such

that

(1) ?i(x)? =?1

(2) ?i(x) − i(y)?2= d(x,y) for all x,y ∈ X.

Proof. We remark first that X is quasihypermetric, by Theorem 3.1

of [10]. Define the semi-inner product space Y by setting Y = M(X)

and (µ | ν) := M(X)µ(X)ν(X) − I(µ,ν) for µ,ν ∈ M(X). Let Y0=

{µ ∈ Y : ?µ? = 0}. Now let H be the completion of the inner product

space Y/Y0, and define i: X → H by

i(x) :=

√2δx+ Y0

for x ∈ X. Now

??i(x)??2

=

2M(X)?1

2for all x ∈ X and

1

=

?1

1

2?δx?2

1

2M(X)

√2δx+ Y0

???

1

√2δx+ Y0

?

=

for all x ∈ X, and

??i(x) − i(y)??2

=

?1

1

2?δx− δy?2

= −1

= d(x,y)

√2(δx− δy) + Y0

???

1

√2(δx− δy) + Y0

?

=

2I(δx− δy,δx− δy)

for all x,y ∈ X. Finally, the image i(X) of X in H is homeomorphic

to X, and therefore separable, and standard arguments show that the

closure of the subspace generated by i(X) is separable.

?

To continue our discussion of metric embeddings, we require the fol-

lowing result, which gives more detailed information than Theorem 4.1

above on the relation between M(X) and the value of M on the finite

subsets of X.

Page 13

DISTANCE GEOMETRY IN QUASIHYPERMETRIC SPACES. III 13

Theorem 4.7. Let (X,d) be a compact quasihypermetric space. Let

(xn)n≥1be any dense sequence in X and write Xn= {x1,...,xn} for

each n ∈ N. Then M(Xn) ↑ M(X) as n → ∞.

Proof. The values M(Xn) are obviously non-decreasing, so that con-

vergence of M(Xn) to M(X) is all that we need to prove. Suppose first

that M(X) < ∞, so that also M(Xn) < ∞ for all n. Applying Theo-

rems 4.11 and 3.1 of [11] to Xnfor each n ≥ 2, we obtain a measure

µn∈ M1(Xn) such that dµn(xi) = M(Xn) for all i such that 1 ≤ i ≤ n.

If n and m are integers with n > m, then we have

?µn− µm?2

= 2I(µn,µm) − I(µn) − I(µm)

= 2µm(dµn) − µn(dµn) − µm(dµm)

= 2M(Xn) − M(Xn) − M(Xm)

= M(Xn) − M(Xm).

Therefore, M(Xm) ≤ M(Xn) = I(µn) ≤ M(X) < ∞ whenever n > m,

and so there exists β ∈ R such that M(Xn) ↑ β as n → ∞. Hence

?µn− µm? → 0 as n,m → ∞. By part (5) of Theorem 5.3 of [10] we

conclude that dµnis a Cauchy sequence in C(X), and hence that there

exists f ∈ C(X) such that dµn→ f in C(X) as n → ∞.

Now fix k ≥ 1, and let n ≥ max(k,2). Since dµn(xk) = M(Xn),

we have dµn(xk) → β as n → ∞, and hence f(xk) = β for all k ≥ 1.

Since xnis a dense sequence in X and f is continuous on X, we have

f(x) = β for all x ∈ X, and hence dµn→ β · 1 in C(X). Thus we

have shown that µnis a d-invariant sequence with value β and that

I(µn) ↑ β. An application of Theorem 4.9 of [11] now gives M(X) = β,

as required.

Now suppose that M(X) = ∞. If M(Xn0) = ∞ for any n0, then

clearly M(Xn) = ∞ for all n ≥ n0, and there is nothing to prove, so

suppose that M(Xn) < ∞ for all n. Fix K > 0. By Theorem 4.1,

there is a finite subset Y = {y1,...,ym} of X such that M(Y ) > K,

and hence a measure µ ∈ M1(Y ) such that I(µ) > K. Write µ =

?m

xni,kwith members chosen from the dense sequence (xn)n≥1such that

xni,k→ yias k → ∞. Then, setting µk =?m

we clearly have µk → µ weak-∗ in M1(X) as k → ∞. Hence, by

Theorem 2.6 of [10] (or Corollary 2.7 of [10]), we have I(µk) → I(µ)

as k → ∞. It follows that for sufficiently large N there exists ν ∈

M1(XN) such that I(ν) > K. Therefore, M(Xn) → ∞, as required.

i=1wiδyifor suitable w1,...,wm∈ R. For each i, pick a sequence

i=1wiδxni,kfor each k,

?

Page 14

14PETER NICKOLAS AND REINHARD WOLF

Theorem 4.8. Let (X,d) be a compact metric space with M(X) < ∞.

Let i: X → H be an S-embedding of X into a Hilbert space H. Then

i(X) lies on a sphere in H of radius r, where M(X) = 2r2.

Proof. As in Theorem 4.7, choose a dense sequence (xn)n≥1in X, write

Xn= {x1,...,xn} for each n ∈ N, and let µn∈ M1(Xn) be a maximal,

and hence d-invariant, measure on Xn. We may assume that xi?= xj

when i ?= j. For any µ ∈ [δx1,δx2,...], the linear span of {δx1,δx2,...},

we have µ =?n

define zµ∈ H by zµ=?n

Note that if µ(X) = 0, then we have?

follows that

?µ?2

n

?

k=1

ℓ=1

k=1βkδxkfor suitable n ∈ N and β1,...,βn∈ R, and we

k=1βki(xk).

kβk = 0, from which it

= −I(µ)

= −

n

?

βkβℓd(xk,xℓ)

= −

n

?

k=1

n

?

ℓ=1

βkβℓ

??i(xk) − i(xℓ)??2

= 2?zµ?2,

2?µ?2.

giving ?zµ?2=

straightforward calculation gives ?zµ− i(x)?2= dµ(x) −1

x ∈ X.

Now (µm− µn)(X) = 0 for all m and n, so we can apply the first

observation above, obtaining ?zµm− zµn?2=

and n. Also, by the proof of Theorem 4.7, the measures µnform a d-

invariant sequence in X, and so we have ?zµm−zµn? → 0 as m,n → ∞.

Hence, as H is complete, there exists z ∈ H such that zµn→ z as

n → ∞. Since µn(X) = 1 for each n ∈ N, we can apply the second

observation above, obtaining ?zµn− i(x)?2= dµn(x) −1

n ∈ N and x ∈ X. Finally, taking limits and using Theorem 4.7, we

have ?z − i(x)?2=1

We conclude our discussion of embeddings by showing that the first

part of Theorem 3.4 generalizes in a natural way to the general compact

case.

1

Also note that if µ(X) = 1, then another

2I(µ) for all

1

2?µm− µn?2for all m

2M(Xn) for all

2M(X) for all x ∈ X, and the result follows.

?

Theorem 4.9. Let (X,d) be a compact quasihypermetric space. Then

the following conditions are equivalent.

(1) M(X) < ∞.

(2) There exists a d-invariant sequence in M1(X).

(3) Some S-embedding of X in the Hilbert space ℓ2lies on a sphere.

Page 15

DISTANCE GEOMETRY IN QUASIHYPERMETRIC SPACES. III15

(4) Every S-embedding of X in the Hilbert space ℓ2lies on a sphere.

Proof. The equivalence of (1) and (2) is given by Corollary 4.10 of [11],

the fact that (1) implies (4) is given by Theorem 4.8, and the fact that

(4) implies (3) is given by an application of Schoenberg’s result as in

the proof of Theorem 3.4.

Suppose that X can be S-embedded on a sphere of radius r in ℓ2.

Clearly every finite subset F of X can then be S-embedded on a sphere

of radius at most r in a suitable euclidean space, and hence satisfies

M(F) ≤ 2r2, by Theorem 3.2. It is now immediate by Theorem 4.1

that M(X) < ∞. Thus, (3) implies (1), completing the proof.

5. M(X) in Finite Spaces

?

In this paper and the earlier paper [11] we have derived several re-

sults about the constant M(X) in a finite metric space X, and have

introduced a number of finite metric spaces or classes of such spaces

as examples and counterexamples. The examples have typically been

constructed so as to have the minimum number of elements consistent

with the phenomenon under discussion.

Our main general result on finite spaces in the present paper so far

has been Theorem 3.4 above, and Theorem 5.3 of [11] was the main

such result in the earlier paper. We reproduce the latter result here for

convenience.

Theorem 5.1 (= Theorem 5.3 of [11]). Let (X,d) be a finite quasihy-

permetric space. Then we have the following.

(1) If X is strictly quasihypermetric, then M(X) < ∞.

(2) If X is not strictly quasihypermetric, then M(X) < ∞ if and

only if there exists no d-invariant measure µ ∈ M0(X) with

value c ?= 0.

In this section, we develop further results about finite spaces, and

in particular settle some of the minimality questions raised by our

examples.

When the space X is finite, the question of the finiteness of M(X)

can be resolved by a straightforward algebraic test, according to the

next result, which also gives an algorithm for the computation of M(X)

when it is finite. We note that Alexander and Stolarsky [1, Theo-

rem 3.3] give a simple algorithm involving the solution of a system of

linear equations for the computation of M(X) when X is a (strictly

quasihypermetric) finite subset of euclidean space.

Theorem 5.2. Let (X = {x1,...,xn},d) with n ≥ 2 be a finite quasi-

hypermetric space. Consider the linear system Dw = 1, where D is

Page 16

16PETER NICKOLAS AND REINHARD WOLF

the distance matrix

length n. Then a solution w = (w1,...,wn)Tto the system exists.

Further,

?n

(2) if

?n

n

?

i=1

?d(xi,xj)?n

i,j=1and 1 is the vector (1,...,1)Tof

(1) if

i=1wi= 0, then M(X) = ∞, and

i=1wi= w0?= 0, then M(X) = 1/w0< ∞, and

(1/w0)wiδxi∈ M1(X)

is a maximal measure on X.

Proof. If M(X) = ∞, part (1) of Theorem 5.1 implies that X is not

strictly quasihypermetric, and then part (2) of Theorem 5.1 implies

that there exists an invariant measure µ ∈ M0(X) with some non-zero

value c. We therefore clearly have a solution w to the linear system.

If M(X) < ∞, then there exists an invariant measure µ ∈ M1(X), by

Theorem 4.11 of [11]. This measure has value M(X) by Theorem 3.1

of [11], and since n ≥ 2 we have M(X) > 0. We therefore again

have a solution to the linear system. Statement (1) now follows from

Theorem 5.2 of [11], and statement (2) from Theorem 3.1 of [11].

?

Remark 5.3. Implicit in the statement and proof of the last theorem

is the fact that?n

the system Dw = 1. Also, since this system always has a solution,

the matrix D is non-singular if and only if the system has exactly one

solution.

i=1wi has the same value for every solution w to

Remark 5.4. If X in the theorem is strictly quasihypermetric, then

the distance matrix D is in fact non-singular. Indeed, D is the natural

matrix representation of the operator T : M(X) → C(X) defined by

T(µ) = dµ for µ ∈ M(X), which is discussed and used extensively

in [10]. If X is strictly quasihypermetric then Theorem 3.6 of [10]

shows that T is an injection (and hence, since X is finite, a bijection),

and so D is non-singular.

Moreover, Theorem 5.8 of [10] implies that if M(X) < ∞, then D is

non-singular only if X is strictly quasihypermetric. Example 5.7 of [11]

provides an example of a 4-point space X which is quasihypermetric

but not strictly quasihypermetric and satisfies M(X) < ∞, and for

which D is therefore singular.

Remark 5.5. A direct calculation using the space presented in Theo-

rem 5.4 of [11] shows that when X is quasihypermetric but not strictly

Page 17

DISTANCE GEOMETRY IN QUASIHYPERMETRIC SPACES. III17

quasihypermetric and has M(X) = ∞, it is possible to have the corre-

sponding distance matrix D non-singular. We present an example to

show that D may also be singular under the same assumptions.

Let X = {x1,x2,x3} have the metric d1with respect to which all non-

zero distances equal 6. Then X is clearly (strictly) quasihypermetric.

Also, the measure µ1=

have M(X) = 4. Let Y = {y1,y2,y3,y4}, where y1,y2,y3,y4are equally

spaced points placed consecutively around a circle of radius

give Y the arc-length metric d2. Using the proof of Corollary 3.3 of [11]

and Example 5.7 of [11], we find that Y is quasihypermetric but not

strictly quasihypermetric, that1

M(Y ) = 2.

Let Z = X ∪ Y and set c = 3. Then defining d: Z × Z → R as in

Theorem 3.5 of [11], we find that Z is quasihypermetric, while Z fails to

be strictly quasihypermetric since Y is not strictly quasihypermetric.

Further, by Theorem 3.6 of [11], the measure −µ1+ µ2∈ M0(Z) is

invariant with value −1, and it follows by Theorem 5.1 that M(Z) = ∞.

Finally, the distance matrix D for Z is singular, since its null space is

the 1-dimensional space spanned by the vector (0,0,0,−1,1,−1,1)T.

1

3(δx1+ δx2+ δx3) is invariant on X, and we

4

π, and

2(δy1+ δy3) is invariant on Y , and that

Our next result gives a systematic account of the relationships that

must hold between the number of points in a finite space and certain

of the metric properties of the space.

First, we recall the following definition, due to Kelly [9]. Let (X,d)

be a metric space. If for all n ∈ N and for all a1,...,an,b1,...,bn+1∈ X

we have

n

?

i=1

n

?

j=1

d(ai,aj) +

n+1

?

i=1

n+1

?

j=1

d(bi,bj) ≤ 2

n

?

i=1

n+1

?

j=1

d(ai,bj),

then (X,d) is said to be a hypermetric space.

Theorem 5.6. The following table gives the necessary relations be-

tween the number of points in a finite space and various metric proper-

ties of the space. (An entry of a dash ‘—’ should be read as ‘sometimes

yes and sometimes no’.)

Page 18

18PETER NICKOLAS AND REINHARD WOLF

number

of points

in space

euclideanhyperme-

tric

quasihy-

permetric

strictly

quasihy-

permetric

M < ∞

≤ 3 yesyes yes yesyes

4— yes yes— yes

≥ 5—————

Proof. The following well established general results (some of which we

have already mentioned here or in [10]) deal with a number of cases

immediately.

(1) By Theorem 3.8 of [1], all compact subsets of euclidean spaces

have M finite.

(2) By Theorem 5.1 of [7], all euclidean spaces are hypermetric.

(3) By Theorem 2 of [8], all hypermetric spaces are quasihyperme-

tric.

(4) By Lemma 1 of [4], all compact subsets of euclidean spaces are

strictly quasihypermetric. (The fact that finite subsets of eu-

clidean space are strictly quasihypermetric was proved in [12].)

The only entries in the table now needing comment are disposed of

(with some redundancy) by the following observations.

(5) Every 4-element metric space is L1-embeddable, by [15] (the

authors are grateful to David Yost for pointing out this fact

and for locating the reference), and therefore hypermetric (see

Remark 3.5 above). (Blumenthal’s four-point theorem [5, The-

orem 52.1] shows independently that such a space is quasihy-

permetric.)

(6) Example 5.7 of [11] gives a 4-element metric space which is not

strictly quasihypermetric (but is hypermetric).

(7) We noted in (5) that every 4-element metric space is hypermet-

ric, and Remark 3.5 outlines the argument that the value of M

must then be finite.

(8) Theorem 5.4 of [11] constructs a 5-element space which is quasi-

hypermetric but not strictly quasihypermetric and has M infi-

nite.

(9) Assouad [2, Proposition 2] constructs a 5-element metric space

which is quasihypermetric but not hypermetric (further infor-

mation is given in Example 5.8 below).

Page 19

DISTANCE GEOMETRY IN QUASIHYPERMETRIC SPACES. III 19

(10) Theorem 3.8 of [11] gives an example of a 5-point space which is

not quasihypermetric. (The optimality of the number 4 in Blu-

menthal’s four-point theorem also corresponds to the existence

of such a space.)

?

A natural question raised by the above results is whether a strictly

quasihypermetric metric space must be hypermetric. We have seen in

part (5) of the proof of Theorem 5.6 that there is no 4-point counterex-

ample, but we present one with 5 points.

Example 5.7. Let X = {x1,x2} and Y = {y1,y2,y3}, and give each

set the discrete metric. If we define Z as in Theorem 3.5 of [11], taking

c =5

ai = xi for i = 1,2 and bj = yj for j = 1,2,3, we find using the

definition of Kelly above that Z is not hypermetric.

8, then it follows that Z is strictly quasihypermetric. But taking

Example 5.8. We show that the 5-element space of Assouad referred

to in part (9) of the proof of Theorem 5.6 is not strictly quasihyper-

metric and has M infinite. The distances in this space are represented

in the obvious way by the entries of the following matrix:

It is easy to check that if we define a measure µ of mass 0 on the space

by using the matrix of respective weights

then we have dµ≡ 2, and the desired conclusions are given by Theo-

rem 5.2 of [11].

0 2 2 5 5

2 0 4 3 3

2 4 0 3 3

5 3 3 0 4

5 3 3 4 0

.

2

−2

−2

1

1

,

References

[1] Ralph Alexander and Kenneth B. Stolarsky, Extremal problems of distance

geometry related to energy integrals, Trans. Amer. Math. Soc. 193 (1974), 1–

31.

[2] Patrice Assouad, Un espace hyperm´ etrique non plongeable dans un espace L1,

C. R. Acad. Sci. Paris S´ er. A-B 285 (1977), A361–A363.

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20 PETER NICKOLAS AND REINHARD WOLF

[3]

, Sur les in´ egalit´ es valides dans L1, European J. Combin. 5 (1984),

99–112.

[4] G¨ oran Bj¨ orck, Distributions of positive mass, which maximize a certain gen-

eralized energy integral, Ark. Mat. 3 (1956), 255–269.

[5] Leonard M. Blumenthal, Theory and applications of distance geometry, Second

edition, Chelsea Publishing Co., New York, 1970.

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323–345.

[7] John B. Kelly, Metric inequalities and symmetric differences, Inequalities, II

(Proc. Second Sympos., U.S. Air Force Acad., Colo., 1967), Academic Press,

New York, 1970, pp. 193–212.

[8]

, Hypermetric spaces and metric transforms, Inequalities, III (Proc.

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memory of Theodore S. Motzkin), Academic Press, New York, 1972, pp. 149–

158.

[9]

, Hypermetric spaces, The geometry of metric and linear spaces (Proc.

Conf., Michigan State Univ., East Lansing, Mich., 1974), Springer, Berlin,

1975, pp. 17–31. Lecture Notes in Math., Vol. 490.

[10] Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric

spaces.I, Bull. Aust. Math. Soc., to appear.

[11]

, Distance geometry in quasihypermetric spaces.II, Math. Nachr., to

appear.

[12] I. J. Schoenberg, On certain metric spaces arising from Euclidean spaces by a

change of metric and their imbedding in Hilbert space, Ann. of Math. (2) 38

(1937), 787–793.

[13]

, Metric spaces and positive definite functions, Trans. Amer. Math. Soc.

44 (1938), 522–536.

[14] H. S. Witsenhausen, Metric inequalities and the zonoid problem, Proc. Amer.

Math. Soc. 40 (1973), 517–520.

[15] Dorothy Wolfe, Imbedding a finite metric set in an N-dimensional Minkowski

space, Nederl. Akad. Wetensch. Proc. Ser. A 70 = Indag. Math. 29 (1967),

136–140.

School of Mathematics and Applied Statistics, University of Wol-

longong, Wollongong, NSW 2522, Australia

E-mail address: peter nickolas@uow.edu.au

Institut f¨ ur Mathematik, Universit¨ at Salzburg, Hellbrunnerstrasse 34,

A-5020 Salzburg, Austria

E-mail address: Reinhard.Wolf@sbg.ac.at