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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