aa r X i v : . [ m a t h . F A ] N ov STRICT p -NEGATIVE TYPE OF A METRIC SPACE HANFENG LI ♭ AND ANTHONY WESTON ♮ Abstract.
Doust and Weston [8] have introduced a new method called en-hanced negative type for calculating a non-trivial lower bound ℘ T on the supre-mal strict p -negative type of any given finite metric tree ( T, d ). In the contextof finite metric trees any such lower bound ℘ T > X, d ) that is knownto have strict p -negative type for some p ≥
0. This allows us to significantlyimprove the lower bounds on the supremal strict p -negative type of finite met-ric trees that were given in [8, Corollary 5.5] and, moreover, leads in to one ofour main results: The supremal p -negative type of a finite metric space cannotbe strict. By way of application we are then able to exhibit large classes offinite metric spaces (such as finite isometric subspaces of Hadamard manifolds)that must have strict p -negative type for some p >
1. We also show that if ametric space (finite or otherwise) has p -negative type for some p >
0, then itmust have strict q -negative type for all q ∈ [0 , p ). This generalizes Schoenberg[27, Theorem 2] and leads to a complete classification of the intervals on whicha metric space may have strict p -negative type. Introduction and Synopsis
The study of positive definite kernels and the related notion of p -negative typemetrics dates back to the early 1900s with some antecedents in the 1800s. A majortheme that emerged was the search for metric characterizations of subsets of Hilbertspace up to isometry. Significant initial results on this classical embedding problemwere obtained by Cayley [6], Menger [21, 22, 23] and Schoenberg [26, 27, 28]. Wenote in particular [28, Theorem 1]: A metric space is isometric to a subset ofHilbert space if and only if it has 2-negative type. This result was spectacularlygeneralized to the category of normed spaces by Bretagnolle et al. [5, Theorem 2]: Areal normed space is linearly isometric to a subspace of some L p -space (1 ≤ p ≤ p -negative type. It remains a prominent question to give acomplete generalization of this result to the setting of non-commutative L p -spaces.See, for example, Junge [15].More recently, difficult questions concerning p -negative type metrics have fig-ured prominently in theoretical computer science. A prime example is the re-cently refuted Goemans-Linial conjecture : Every metric space of 1-negative typebi-Lipschitz embeds into some L -space. Although this conjecture clearly holds fornormed spaces by [5, Theorem 2] (with p = 1), it is not true for arbitrary metricspaces. This was shown by Khot and Vishnoi [18]. Subsequently, Lee and Naor [19] Mathematics Subject Classification.
Key words and phrases.
Finite metric spaces, strict p -negative type, generalized roundness. ♭ Partially supported by NSF grant DMS-0701414. ♮ Partially supported by internal research grants at Canisius College. have shown that there is no metric version of [5, Theorem 2] (modulo bi-Lipschitzembeddings) for any p ∈ [1 , p ∈ [1 ,
2) there is a metric space (
X, d ) of p -negative type which does notbi-Lipschitz embed into any L p -space.The related notion of strict p -negative type has been studied rather less wellthan its classical counterpart and most known results deal with the case p = 1.The present work is motivated by functional analytic questions that arise naturallyfrom the papers of Hjorth et al. [13, 14]. Both [13] and [14] focus on examplesand properties of finite metric spaces of strict 1-negative type. One theme of thesepapers is to determine global geometric properties of finite metric spaces of strict1-negative type. As an example we mention [14, Theorem 3.9]: If a finite metricspace is of strict 1-negative type, then it has a unique ∞ -extender. It is also naturalto ask for conditions on a finite metric space which will guarantee that it has strict1-negative type. One such result is [14, Theorem 5.2]: If a finite metric space ishypermetric and regular, then it is of strict 1-negative type.The theme of this paper is to determine basic properties of strict p -negativemetrics for all p ≥
0. In particular, we aim to move beyond the familiar case p = 1,thereby setting up the rudiments of a basic theory of strict p -negative type metrics.Section 2 is dedicated to a review of the salient features of generalized roundness,negative type, and strict negative type. In Definition 2.7 we recall the notion ofthe (normalized) p -negative type gap Γ pX of a metric space ( X, d ). This parameterwas recently introduced by Doust and Weston [8, 9] in order to obtain non-triviallower bounds on the maximal p -negative of finite metric trees. Basic properties ofΓ pX will play a vital rˆole in our computations in Section 3.The observation is made in [8, Theorem 5.2] that if the p -negative type gap Γ pX of a finite metric space ( X, d ) is positive for some p ≥
0, then (
X, d ) must havestrict q -negative type on some interval of the form [ p, p + ζ ) where ζ >
0. However,the authors only provide an explicit value for ζ in the case p = 1. Letting n = | X | ,the value of ζ given in this case is O (1 /n ). (See [8, Theorem 5.1].) The purpose ofSection 3 is to give a precise quantitative version of [8, Theorem 5.2] which yieldssignificantly improved values of ζ for all p ≥
0. In fact, for each p ≥
0, our valueof ζ is O (1). The precise statement of this result is given in Theorem 3.3. Byway of application, Theorem 3.3 leads to significantly improved lower bounds onthe maximal p -negative type of finite metric trees. These are given in Corollary3.4. Then in Remark 3.5 we point out that the estimates given in Corollary 3.4 arereasonably sharp for finite metric trees that resemble stars. This suggests there islittle room for improvement in the statement of Theorem 3.3 (in general).In Section 4 we use Theorem 3.3 and an elementary compactness argument toderive a key result of this paper: The supremal p -negative type of a finite metricspace cannot be strict. This is done in Corollary 4.3 to Theorem 4.1. Using knownresults we are then able to exhibit large classes of finite metric spaces, all of whichmust have strict p -negative type for some p >
1. For example, any finite isometricsubspace of a Hadamard manifold must have strict p -negative type for some p > p -negative type for some p >
0, then it must havestrict q -negative type for all q ∈ [0 , p ). This allows us to precisely codify the TRICT p -NEGATIVE TYPE OF A METRIC SPACE 3 types of intervals on which a metric space may have strict p -negative type. Itis interesting to note that finite metric spaces behave quite differently to infinitemetric spaces in this respect. These differences are highlighted in Theorems 5.5 and5.8. Understanding how strict negative type behaves on intervals leads to furtherexamples of metric spaces that have non-trivial strict p -negative type. We thenconclude the paper with the observation in Remark 5.13 that Theorems 3.3, 4.1and 5.4 (as well as several of our corollaries) actually hold more generally for finitesemi-metric spaces. This is because we do not use the triangle inequality at anypoint in our definitions or proofs.Throughout this paper the set of natural numbers N is taken to consist of allpositive integers and sums indexed over the empty set are always taken to be zero.Given a real number x , we are using ⌊ x ⌋ to denote the largest integer that does notexceed x , and ⌈ x ⌉ to denote the smallest integer which is not less than x .2. A framework for ordinary and strict p -negative type We begin by recalling some theoretical features of (strict) p -negative type andits relationship to (strict) generalized roundness. More detailed accounts may befound in the work of Benyamini and Lindenstrauss [3], Deza and Laurent [7], Pras-sidis and Weston [25], and Wells and Williams [29]. These works emphasize theinterplay between the classical p -negative type inequalities and isometric, Lipschitzor uniform embeddings. They also indicate applications to more contemporary ar-eas of interest such as theoretical computer science. One of the most importantresults for our purposes is the equivalence of (strict) p -negative type and (strict)generalized roundness p . These equivalences are described in Theorem 2.5. Definition 2.1.
Let p ≥ X, d ) be a metric space. Then:(a) (
X, d ) has p - negative type if and only if for all natural numbers k ≥
2, allfinite subsets { x , . . . , x k } ⊆ X , and all choices of real numbers η , . . . , η k with η + · · · + η k = 0, we have: X ≤ i,j ≤ k d ( x i , x j ) p η i η j ≤ . (1)(b) ( X, d ) has strict p - negative type if and only if it has p -negative type and theassociated inequalities (1) are all strict except in the trivial case ( η , . . . , η k )= (0 , . . . , Remark . Every metric space obviously has strict 0-negative type. It is also thecase that every finite metric space has strict p -negative type for some p >
0. Thisfollows from Weston [30, Theorem 4.3], Theorem 2.5 (a) and Theorem 5.4.It is possible to reformulate both ordinary and strict p -negative type in terms ofan invariant known as generalized roundness from the uniform theory of Banachspaces. Generalized roundness was introduced by Enflo [11] in order to solve (inthe negative) Smirnov’s Problem : Is every separable metric space uniformly home-omorphic to a subset of Hilbert space? The analog of this problem for coarseembeddings was later raised by Gromov [12] and solved negatively by Dranishnikovet al. [10]. Prior to introducing generalized roundness in Definition 2.4 (a) we shalldevelop some intermediate technical notions in order to streamline the expositionin the remainder of this paper.
HANFENG LI AND ANTHONY WESTON
Definition 2.3.
Let s, t be arbitrary natural numbers and let X be any set.(a) An ( s, t )- simplex in X is an ( s + t )-vector ( a , . . . , a s , b , . . . , b t ) ∈ X s + t consisting of s + t pairwise distinct coordinates a , . . . , a s , b , . . . , b t ∈ X .Such a simplex will be denoted by D = [ a j ; b i ] s,t .(b) A load vector for an ( s, t )-simplex D = [ a j ; b i ] s,t in X is an arbitrary vector ~ω = ( m , . . . m s , n , . . . , n t ) ∈ R s + t + that assigns a positive weight m j > n i > a j or b i of D , respectively.(c) A loaded ( s, t )- simplex in X consists of an ( s, t )-simplex D = [ a j ; b i ] s,t in X together with a load vector ~ω = ( m , . . . , m s , n , . . . , n t ) for D . Such aloaded simplex will be denoted by D ( ~ω ) or [ a j ( m j ); b i ( n i )] s,t as the needarises.(d) A normalized ( s, t )- simplex in X is a loaded ( s, t )-simplex D ( ~ω ) in X whoseload vector ~ω = ( m , . . . , m s , n , . . . , n t ) satisfies the two normalizations: m + · · · + m s = 1 = n + · · · n t . Such a vector ~ω will be called a normalized load vector for D .Rather than giving the original definition of generalized roundness p from [11], weshall present an equivalent reformulation in Definition 2.4 (a) that is due to Lennardet al. [20] and Weston [30]. Definition 2.4.
Let p ≥ X, d ) be a metric space. Then:(a) (
X, d ) has generalized roundness p if and only if for all s, t ∈ N and allnormalized ( s, t )-simplices D ( ~ω ) = [ a j ( m j ); b i ( n i )] s,t in X we have: X ≤ j Let p ≥ X, d ) be a metric space. Let s, t be naturalnumbers and D = [ a j ; b i ] s,t be an ( s, t )-simplex in X . Denote by N s,t the setof all normalized load vectors ~ω = ( m , . . . , m s , n , . . . , n t ) ⊂ R s + t + for D . Thenthe (normalized) p - negative type simplex gap of D is defined to be the function TRICT p -NEGATIVE TYPE OF A METRIC SPACE 5 γ pD : N s,t → R where γ pD ( ~ω ) = s,t X j,i =1 m j n i d ( a j , b i ) p − X ≤ j Let p ≥ 0. Let ( X, d ) be a metric space with p -negative type.We define the (normalized) p - negative type gap of ( X, d ) to be the non-negativequantity Γ pX = inf D ( ~ω ) γ pD ( ~ω )where the infimum is taken over all normalized ( s, t )-simplices D ( ~ω ) in X .Recall that a finite metric tree is a finite connected graph that has no cycles,endowed with an edge weighted path metric. Hjorth et al. [14] have shown thatfinite metric trees have strict 1-negative type. Therefore it makes sense to tryto compute the 1-negative type gap of any given finite metric tree. Indeed, avery succinct formula was derived in [8, Corollary 4.13]. However, a modicum ofadditional notation is necessary before stating this result. The set of all edges ina metric tree ( T, d ), considered as unordered pairs, will be denoted E ( T ), and themetric length d ( x, y ) of any given edge e = ( x, y ) ∈ E ( T ) will be denoted | e | . Theorem 2.8 (Doust and Weston [8]) . Let ( T, d ) be a finite metric tree. Then the(normalized) -negative type gap Γ = Γ T of ( T, d ) is given by the following formula: Γ = ( X e ∈ E ( T ) | e | − ) − . In particular, Γ > . In the remaining sections of this paper we shall show how the notions, equivalencesand results of this section may be used to infer some basic properties of metrics ofstrict p -negative metrics for general values of p ≥ A quantitative lower bound on supremal strict p -negative type The observation is made in [8, Theorem 5.2] that if the p -negative type gap Γ pX of a finite metric space ( X, d ) is positive for some p ≥ 0, then ( X, d ) must havestrict q -negative type on some interval of the form [ p, p + ζ ) where ζ > 0. However,the authors only provide an explicit value for ζ in the case p = 1. Letting n = | X | ,the value of ζ given in this case is O (1 /n ). (See [8, Theorem 5.1].) The purposeof the present section is to give a precise quantitative version of [8, Theorem 5.2]which yields significantly improved values of ζ for all p ≥ 0. In fact, for each p ≥ ζ is O (1). The precise statement of this result is given in Theorem 3.3.As an application we obtain significantly improved lower bounds on the maximal p -negative type of finite metric trees. These are stated in Corollary 3.4. Then in HANFENG LI AND ANTHONY WESTON Remark 3.5 we point out that the estimates given in Corollary 3.4 are actually closeto best possible for finite metric trees that resemble stars. This suggests there islittle room for improvement in the statement of Theorem 3.3, the main result ofthis section.The proof of Theorem 3.3 is facilitated by the following two technical lemmaswhich are easily realized using basic calculus or by simple combinatorial arguments.The proofs of these lemmas are therefore omitted. Lemma 3.1. Let s ∈ N . If s real variables ℓ , . . . , ℓ s > are subject to theconstraint ℓ + · · · + ℓ s = 1 , then the expression X k Let s, t ∈ N and let m = s + t . Then (cid:18) − s (cid:19) + 12 (cid:18) − t (cid:19) ≤ − (cid:18) ⌊ m ⌋ + 1 ⌈ m ⌉ (cid:19) . Moreover, the function γ ( m ) = 1 − (cid:16) ⌊ m ⌋ + ⌈ m ⌉ (cid:17) increases strictly as m increases. We will continue to use the notation γ ( m ) = 1 − · (cid:0) ⌊ m ⌋ − + ⌈ m ⌉ − (cid:1) introducedin the preceding lemma throughout the remainder of this section as it allows theefficient statement and succinct proof of certain key formulas such as Theorem 3.3.The following basic notions are also relevant to the proof of Theorem 3.3. Let( X, d ) be a metric space. If d ( x, y ) = 1 for all x = y , then d is called the discretemetric on X . The metric diameter of ( X, d ) is given by the quantity diam X =sup { d ( x, y ) | x, y ∈ X } . Provided | X | < ∞ , the scaled metric diameter of ( X, d ) isgiven by the ratio D X = (diam X ) / min { d ( x, y ) | x = y } . Theorem 3.3. Let ( X, d ) be a finite metric space with cardinality n = | X | ≥ and let p ≥ . If the p -negative type gap Γ pX of ( X, d ) is positive, then ( X, d ) has q -negative type for all q ∈ [ p, p + ζ ] where ζ = ln (cid:18) Γ pX (diam X ) p · γ ( n ) (cid:19) ln D X . Moreover, ( X, d ) has strict q -negative type for all q ∈ [ p, p + ζ ) . In particular, p + ζ provides a lower bound on the supremal (strict) q -negative type of ( X, d ) .Proof. For notational ease we set Γ = Γ pX and D = D X throughout this proof. Wemay assume that the metric d is not a positive multiple of the discrete metric on X . Otherwise, ( X, d ) would have strict q -negative type for all q ≥ 0. Hence D > p -negative type, we may assume that min { d ( x, y ) | x = y } = 1. Thismeans that D is now the diameter of our rescaled metric space (which we willcontinue to denote by ( X, d )). Moreover, for all ℓ = d ( x, y ) = 0 and all ζ > 0, wehave ℓ p + ζ − ℓ p ≤ D p + ζ − D p . This is because, for any fixed ζ > 0, the function f ( x ) = x p + ζ − x p is increasing on the interval [1 , ∞ ). This inequality will be usedin the derivation of (5) below. TRICT p -NEGATIVE TYPE OF A METRIC SPACE 7 Consider an arbitrary normalized ( s, t )-simplex D = [ a j ( m j ); b i ( n i )] s,t in X .Necessarily, m = s + t ≤ n . For any given r ≥ 0, let L ( r ) = X j 0. Therefore we only need to concentrateon the first inequality of (4). First of all, notice that L ( p + ζ ) − L ( p ) = X j Recall that the ordinary path metric on a finite tree T assigns length one to eachedge in the tree (with all other distances determined geodesically). With this inmind, we see that Theorem 3.3 provides a significant improvement of the estimategiven in [8, Corollary 5.5]. Corollary 3.4. Let T be a finite tree on n = | T | ≥ vertices that is endowed withthe ordinary path metric d . Let D denote the metric diameter of the resulting finitemetric tree ( T, d ) . Let ℘ T denote the maximal p -negative type of ( T, d ) . Then: ℘ T ≥ (cid:26) ln (cid:18) D · ( n − · γ ( n ) (cid:19). ln D (cid:27) . (7) Proof. By Theorem 2.8, Γ T = n − . Now apply Theorem 3.3 with p = 1. (cid:3) Remark . The lower bound on ℘ T given in the statement of Corollary 3.4 isbasically of the correct order of magnitude when D = 2. To see this, first of allnotice that if n > D = 2, then (7) in Corollary 3.4 simplifies to give: ℘ T ≥ (cid:26) ln (cid:18) n n − n − (cid:19). ln 2 (cid:27) . However, if T denotes a star with n − ℘ T = 1 + (cid:26) ln (cid:18) n − (cid:19). ln 2 (cid:27) . Supremal p -negative type of a finite metric space cannot be strict If the p -negative type gap Γ pX of a metric space ( X, d ) is positive then ( X, d )clearly has strict p -negative type. It is interesting to ask to what extent — ifany — the converse of this statement is true. Our next result points out that theconverse statement is always true in the case of finite metric spaces. By way of anotable contrast, [8, Theorem 5.7] shows that there exist infinite metric trees ( X, d )of strict 1-negative type with 1-negative type gap Γ X = 0. Theorem 4.1. Let p ≥ and let ( X, d ) be a finite metric space. Then ( X, d ) hasstrict p -negative type if and only if Γ pX > .Proof. Let p ≥ X, d ) is a finite metric space with strict p -negative type. ByTheorem 2.5, γ pD ( ~ω ) > s, t )-simplex D ( ~ω ) ⊆ X . Referringback to Definitions 2.3 and 2.6 we further note that we may assume that each such p -negative type simplex gap γ pD is defined on the compact set N s,t ⊂ R s + t and ispositive at each point of N s,t . Thereforemin (cid:26) γ pD ( ~ω ) | ~ω ∈ N s,t (cid:27) > s, t )-simplex D in X . But as | X | < ∞ the number of distinct ( s, t )-simplexes D that can be formed from X must be finite. Thus the p -negative typegap Γ pX is seen to be the minimum of finitely many positive quantities. As such weobtain the desired result: Γ pX > (cid:3) TRICT p -NEGATIVE TYPE OF A METRIC SPACE 9 Corollary 4.2. Let p ≥ and let ( X, d ) be a finite metric space. If ( X, d ) hasstrict p -negative type, then ( X, d ) must have strict q -negative type for some intervalof values q ∈ [ p, p + ζ ) , ζ > .Proof. By Theorem 4.1, Γ = Γ pX > 0. Now apply Theorem 3.3. (cid:3) As an immediate consequence of Corollary 4.2 we obtain one of the main results ofthis paper. Corollary 4.3. The supremal p -negative type of a finite metric space cannot bestrict. Moreover, since p -negative type holds on closed intervals, we therefore obtain aninteresting case of equality in the classical negative type inequalities as a directconsequence of Corollary 4.3. Corollary 4.4. Let ( X, d ) be a finite metric space. Let ℘ denote the supremal p -negative type of ( X, d ) . If ℘ < ∞ then there exists a normalized ( s, t ) -simplex D ( ~ω ) = [ a j ( m j ); b i ( n i )] s,t in X such that γ ℘D ( ~ω ) = 0 . In other words, we obtain: X ≤ j The following finite metric spaces all have strict q -negative typefor some interval of values q ∈ [1 , ζ ) (where ζ > depends upon the particularspace): (a) Any three-point metric space. (b) Any finite metric tree. (c) Any finite isometric subspace of a k -sphere S k (endowed with the usualgeodesic metric) that contains at most one pair of antipodal points. (d) Any finite isometric subspace of the hyperbolic space H k R (or H k C ). (e) Any finite isometric subspace of a Hadamard manifold.Proof. All of the above finite metric spaces have strict p -negative type for p = 1 byresults given in [13] and [14]. We may therefore apply Corollary 4.2 en masse . (cid:3) Range of strict p -negative type It is a classical result of Schoenberg [27, Theorem 2] that p -negative type holdson closed intervals. More precisely, the set of all values of p for which a given metricspace ( X, d ) has p -negative type is always an interval of the form [0 , ℘ ] or [0 , ∞ ).Included here is the possibility that ℘ = 0, in which case the interval degeneratesto { } . Examples of Enflo [11] in tandem with Theorem 2.5 (a) imply that all suchintervals (degenerate or otherwise) can occur. Moreover, for intervals of the form[0 , ℘ ] with ℘ > 0, the examples given in [11, Section 1] are finite metric spaces. Inthe case of the degenerate interval { } the situation is slightly more delicate. Itfollows from [11, Theorem 2.1] and Theorem 2.5 (a) that the Banach space C [0 , p -negative type for any p > 0. In Theorems 5.4, 5.5 and 5.8 weprovide strict versions of [27, Theorem 2]. These theorems allow us to preciselycodify the types of intervals on which a metric space may have strict p -negative type. It is interesting to note that finite metric spaces behave quite differently toinfinite metric spaces in this respect. Theorems 5.5 and 5.8 highlight this point.In order to proceed we must first briefly recall some basic facts about kernelsof positive type and kernels conditionally of negative type. (In some importantrespects we are following Nowak [24, Sections 2–4].) Definition 5.1. Let X be a topological space.(a) A kernel of positive type on X is a continuous function Φ : X × X → C such that for any n ∈ N , any elements x , . . . , x n ∈ X , and any complexnumbers η , . . . , η n we have: X ≤ i,j ≤ n Φ( x i , x j ) η i η j ≥ . (b) A kernel conditionally of negative type on X is a continuous function Ψ : X × X → R with the following three properties:(1) Ψ( x, x ) = 0 for all x ∈ X ,(2) Ψ( x, y ) = Ψ( y, x ) for all x, y ∈ X , and(3) for any n ∈ N , any x , . . . , x n ∈ X , and any real numbers η , . . . , η n with η + · · · + η n = 0 we have: X ≤ i,j ≤ n Ψ( x i , x j ) η i η j ≤ . The following fundamental relationship between kernels of positive type and kernelsconditionally of negative type was given by Schoenberg [28]. For a short proof ofthis theorem we refer the reader to Bekka et al. [1, Theorem C.3.2]. Theorem 5.2 (Schoenberg [28]) . Let X be a topological space and Ψ : X × X → R be a continuous kernel on X such that Ψ( x, x ) = 0 and Ψ( x, y ) = Ψ( y, x ) for all x, y ∈ X . Then Ψ is conditionally of negative type if and only if the kernel Φ = e − t Ψ is of positive type for every t ≥ . Our proof of Theorem 5.4 makes use of the following identity. An explanation ofthis identity may be found in the proof of Corollary 3.2.10 in Berg et al. [2]. Lemma 5.3. For each α ∈ (0 , there exists a constant c α > such that x α = c α ∞ Z (1 − e − tx ) t − α − dt for all x ≥ . Theorem 5.4. Let ( X, d ) be a metric space. If ( X, d ) has p -negative type for some p > , then it must have strict q -negative type for all q such that ≤ q < p .Proof. Every metric space has strict 0-negative type. So we may assume that q > 0. Since ( X, d ) has p -negative type, the function Ψ : X × X → R defined byΨ( x, y ) = d ( x, y ) p is conditionally of negative type. Hence, by Theorem 5.2, thefunction e − t Ψ : X × X → C is of positive type for every t ≥ x , . . . , x n ( n ≥ 2) be distinct points in X and let η , . . . , η n be real numbers,not all zero, such that P j η j = 0. We need to show that P i,j d ( x i , x j ) q η i η j < t ≥ 0, set f ( t ) = X i,j (1 − e − td ( x i ,x j ) p ) η i η j . TRICT p -NEGATIVE TYPE OF A METRIC SPACE 11 Then f ( t ) = X i,j η i η j − X i,j e − td ( x i ,x j ) p η i η j = (cid:18)X j η j (cid:19) − X i,j e − td ( x i ,x j ) p η i η j = − X i,j e − td ( x i ,x j ) p η i η j ≤ t ≥ 0. When t → ∞ , one has f ( t ) → − P j η j < 0. Thus f ( t ) < t sufficiently large. Set α = q/p . By applying Lemma 5.3 to x = d ( x i , x j ) p , one gets X i,j d ( x i , x j ) q η i η j = X i,j (cid:18) c α Z ∞ (1 − e − td ( x i ,x j ) p ) t − α − dt (cid:19) η i η j = c α Z ∞ f ( t ) t − α − dt < , as desired. (cid:3) As an immediate consequence of Corollary 4.3, Theorem 5.4, [30, Theorem 4.3] andthe examples given in [11, Section 1] we obtain the following theorem. Theorem 5.5. Let ( X, d ) be a finite metric space. The set of all values of p forwhich ( X, d ) has strict p -negative type is always an interval of the form [0 , ℘ ) , with ℘ > , or [0 , ∞ ) . Moreover, all such intervals can occur. By way of marked contrast with Theorem 5.5, we note that (a) the set of all valuesof p for which the Banach space C [0 , 1] has strict p -negative type is the degenerateinterval { } (this follows from [11, Theorem 2.1] and Theorem 2.5), and (b) thesupremal p -negative type of an infinite metric space may or may not be strict. Forexample, in [8, Theorem 5.7] the authors construct an infinite metric tree that hasstrict p -negative type if and only if p ∈ [0 , ℓ hasstrict p -negative type if and only if p ∈ [0 , ℘ > X, d ) that has strict p -negative type ifand only if p ∈ [0 , ℘ ]. The following definition introduces the relevant spaces. Definition 5.6. Let ℘ > 0. Let ( ℘ k ) be a strictly decreasing sequence of realnumbers that converges to ℘ . Let n be a natural number that satisfies the condition: (cid:18) − n (cid:19) /℘ ≥ / . Let Y be the union of a sequence of pairwise disjoint sets ( Y , Y , Y , . . . ) such that | Y k | = n for all k ∈ N . Let Z be the union of a sequence of pairwise disjoint sets( Z , Z , Z , . . . ) such that | Z k | = n for all k ∈ N and Y ∩ Z = ∅ . Set X = Y ∪ Z .We metrize X in the following way: d ( y, z ) = (cid:18) − n (cid:19) /℘ k if y ∈ Y k and z ∈ Z k for some k ∈ N . All other non-zero distances in the space aretaken to be one. We call ( X, d ) an Enflo ℘ -space .Notice that in Definition 5.6 the condition placed on n ensures that ( X, d ) reallyis a metric space. Moreover, from the examples noted in [11, Section 1], togetherwith Theorem 2.5 (a), it follows that each subspace Y k ∪ Z k of an Enflo ℘ -space ( X, d ) has maximal p -negative type exactly equal to ℘ k . In order to proceed weneed to develop a slightly stronger statement about certain subspaces of ( X, d ). Lemma 5.7. Let ℘ > . Let ( X, d ) be an Enflo ℘ -space as in Definition 5.6. Foreach m ∈ N , the subspace X m = S { Y k ∪ Z k | ≤ k ≤ m } of ( X, d ) has ℘ m -negativetype. In fact, ℘ m is the maximal p -negative type of the subspace X m .Proof. Let D be a given normalized simplex in X m . Without loss of generality, wemay assume that D = [ y k,j ( α k,j ) , z k,j ( β k,j ); ¯ y k,j ( γ k,j ) , ¯ z k,j ( η k,j )], where y k,j , ¯ y k,j ∈ Y k and z k,j , ¯ z k,j ∈ Z k . In other words, our simplex has the points y k,j , z k,j on oneside and the remaining points ¯ y k,j , ¯ z k,j on the other side (with weights as indicated).Let L ( ℘ m ) and R ( ℘ m ) denote the left side and right side of (2) computed relativeto the simplex D with exponent ℘ m (respectively). By setting α k = P j α k,j , β k = P j β k,j , γ k = P j γ k,j and η k = P j η k,j we may then compute L ( ℘ m ) and R ( ℘ m ): L ( ℘ m ) = X k ( α k β k + γ k η k ) (cid:26)(cid:18) − n (cid:19) ℘ m /℘ k − (cid:27) +1 − X k (cid:18)X j α k,j + X j β k,j + X j γ k,j + X j η k,j (cid:19) , (8)while R ( ℘ m ) = X k ( α k η k + γ k β k ) (cid:26)(cid:18) − n (cid:19) ℘ m /℘ k − (cid:27) +1 . (9)If we let s k denote the number of points y k,j in Y k and let t k denote the numberof points ¯ y k,j in Y k , then it follows that we have: X j α k,j + X j γ k,j ≥ α k s k + γ k t k ≥ α k + γ k n . Similarly, X j β k,j + X j η k,j ≥ β k + η k n . As a result, comparing the expressions (8) and (9), we see that it will follow that L ( ℘ m ) ≤ R ( ℘ m ) provided that we can establish the following non-linear inequality: X k (cid:26) ( α k β k + γ k η k ) − ( α k η k + γ k β k ) (cid:27)(cid:26)(cid:18) − n (cid:19) ℘ m /℘ k − (cid:27) ≤ X k α k + γ k + β k + η k n . (10) TRICT p -NEGATIVE TYPE OF A METRIC SPACE 13 We claim that (10) holds term by term. That is to say, (cid:26) ( α k β k + γ k η k ) − ( α k η k + γ k β k ) (cid:27)(cid:26)(cid:18) − n (cid:19) ℘ m /℘ k − (cid:27) ≤ · α k + γ k + β k + η k n , (11)for each k . In fact, for each k , (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) ( α k β k + γ k η k ) − ( α k η k + γ k β k ) (cid:27)(cid:26)(cid:18) − n (cid:19) ℘ m /℘ k − (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max (cid:0) α k β k + γ k η k , α k η k + γ k β k (cid:1) · n ≤ · α k + β k + γ k + η k n . As (11) implies (10) we conclude that L ( ℘ m ) ≤ R ( ℘ m ). Thus X m has maximal p -negative type at least ℘ m . However the subspace Y m ∪ Z m of X m has maximal p -negative type ℘ m by [11, Section 1] and Theorem 2.5 (a). We conclude that themaximal p -negative type of X m is ℘ m . (cid:3) Theorem 5.8. Let ℘ > . Let ( X, d ) be an Enflo ℘ -space as in Definition 5.6.Then ( X, d ) has strict p -negative type if and only if p ∈ [0 , ℘ ] .Proof. For each k the subspace Y k ∪ Z k of ( X, d ) has maximal p -negative type ℘ k .And since ℘ k ց ℘ as k → ∞ it follows that ( X, d ) does not have p -negative typefor any p > ℘ .However, each subspace X m = Y m ∪ Z m of ( X, d ) has ℘ m -negative type byLemma 5.7. By Theorem 5.4, X m has strict p -negative type for all p ∈ [0 , ℘ m ).Thus ( X, d ) has strict p -negative type for all p ∈ [0 , ℘ ] as asserted. (cid:3) We conclude this paper with some final applications of Theorem 5.4. Recall thatthe maximal q -negative type of certain classical (quasi-) Banach spaces has beencomputed explicitly. For example, suppose 0 < p ≤ µ is a non-trivialpositive measure, then the maximal q -negative type of L p ( µ ) is simply p . A shortproof of this result, which is due to Schoenberg [28] for 1 ≤ p ≤ 2, may be found in[20, Corollary 2.6 (a)]. Theorem 5.4 therefore applies as follows. Corollary 5.9. Let < p ≤ and let µ be a positive measure. Then any metricspace ( X, d ) which is isometric to a subset of L p ( µ ) must have strict q -negative typefor all q ∈ [0 , p ) . Corollary 4.4 and Theorem 5.4 combine to provide the following characterizationof the supremal p -negative type of a finite metric space in terms of zeros of thesimplex gap functions γ qD . Corollary 5.10. If the supremal p -negative type ℘ of a finite metric space ( X, d ) is finite, then: ℘ = min { q | q > and γ qD ( ~ω ) = 0 for some normalized (s,t)-simplex D ( ~ω ) ⊆ X } . In certain instances Theorem 5.4 provides a second description of the maximal p -negative type of a metric space. Corollary 5.11. Let ℘ > . If a metric space ( X, d ) has ℘ -negative type but notstrict ℘ -negative type, then ℘ is the maximal p -negative type of ( X, d ) . It follows from Kelly [16, 17] that any k -sphere S k endowed with the usual geodesicmetric is ℓ -embeddable and therefore of 1-negative type. On the other hand,Hjorth et al. [14, Theorem 9.1] have shown that a finite isometric subspace ( X, d )of a k -sphere S k is of strict 1-negative type if and only if X contains at most one pairof antipodal points. These comments and Corollary 5.11 imply the next corollary. Corollary 5.12. A finite isometric subspace ( X, d ) of a k -sphere S k has maximal p -negative type = 1 if and only if X contains at least two pairs of antipodal points. Recall, following Blumenthal [4], that a semi-metric space is required to satisfy allof the axioms of a metric space except (possibly) the triangle inequality. Remark . In closing we note that Theorems 3.3, 4.1 and 5.4 hold (more gener-ally) for all finite semi-metric spaces ( X, d ). The same goes for Corollaries 4.2, 4.3,4.4, 5.10 and 5.11. This is because the triangle inequality has played no rˆole in anyof the definitions or computations of this paper. 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