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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
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[2] Patrice Assouad, Un espace hyperm´ etrique non plongeable dans un espace L1,
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20 PETER NICKOLAS AND REINHARD WOLF
[3]
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, Distance geometry in quasihypermetric spaces.II, Math. Nachr., to
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[13]
, Metric spaces and positive definite functions, Trans. Amer. Math. Soc.
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[14] H. S. Witsenhausen, Metric inequalities and the zonoid problem, Proc. Amer.
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[15] Dorothy Wolfe, Imbedding a finite metric set in an N-dimensional Minkowski
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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