arXiv subjects

Anthony Weston

Publications and source records attributed to Anthony Weston.

16 recordsLinked to original sources

arXiv2026

Squaring the circle: embedding $S^1$ in $\ell_1$

Let $(S^1,\delta)$ be the unit circle endowed with the arc length metric. This paper concerns the question of which subsets of $(S^1,\delta)$ can be embedded isometrically into the sequence space $\ell_1$. It is well known that every finite subset of $S^1$ admits such an embedding, but the situation for infinite subsets, even including the whole circle, has perhaps been obscured by conflicting terminology in the literature. It is worth noting that the circle avoids the classical obstructions to isometric embeddability, since it is both of negative type and hypermetric. In this paper we show that if $X \subseteq S^{1}$ is a closed set and the Lebesgue measure of $X \cap X^{\ast}$ is positive, where $X^{\ast}$ is the antipodal set of $X$, then it is impossible to isometrically embed $(X, \delta)$ in $\ell_{1}$. As a result, no subset of $S^{1}$ with Lebesgue measure greater than $\pi$ can be isometrically embedded in $\ell_1$. Conversely, we show that any closed subset of $S^1$ whose intersection with any half-circle has measure zero can be isometrically embedded in $\ell_1$. These results have several consequences. They imply that the classical Banach space $L_1[0, 1]$, considered purely as a metric space, does not isometrically embed in $\ell_{1}$. Secondly, they yield a simple proof that a metric graph $(M, d)$ embeds isometrically in $\ell_1$ if and only if it is a tree. In contrast to $S^{1}$, we show that some related metric spaces, such as the cylinder and the flat torus, contain finite subsets that cannot be isometrically embedded in $\ell_1$.

math.FA↗

arXiv2024

Beyond trees: the metric geometry of subsets of weighted Hamming cubes

Associated to any finite metric space are a large number of objects and quantities which provide some degree of structural or geometric information about the space. In this paper we show that in the setting of subsets of weighted Hamming cubes there are unexpected relationships between many of these quantities. We obtain in particular formulas for the determinant of the distance matrix, the $M$-constant and the cofactor sum for such spaces. In general, these types of results offer valuable insights into the combinatorial optimization of certain constrained quadratic forms on finite metric spaces. A key focus in this context are embedding properties of negative type metrics, which play a prominent role in addressing important questions like the sparsest cut problem in graph theory. The current work extends previous results for unweighted metric trees, and more generally, for subsets of standard Hamming cubes, as well as results for weighted metric trees. Finally we consider polygonal equalities in these spaces, giving a complete description of the nontrivial $1$-polygonal equalities that can arise in weighted Hamming cubes.

math.FA↗

arXiv2024

Polygonal equalities and $p$-negative type

Nontrivial $p$-polygonal equalities impose certain conditions on the geometry of a metric space $(X,d)$ and so it is of interest to be able to identify the values of $p \in [0,\infty)$ for which such equalities exist. Following work of Li and Weston, Kelleher, Miller, Osborn and Weston established that if a metric space $(X,d)$ is of $p$-negative type, then $(X,d)$ admits no nontrivial $p$-polygonal equalities if and only if it is of strict $p$-negative type. In this note we remove the underlying premise of $p$-negative type from this theorem. As an application we show that the set of all $p$ for which a finite metric space $(X,d)$ admits a nontrivial $p$-polygonal equality is always a closed interval of the form $[\wp, \infty)$, where $\wp > 0$, or the empty set. It follows that for each $q \not= 2$, the Schatten $q$-class $\mathcal{C}_{q}$ admits a nontrivial $p$-polygonal equality for each $p > 0$. Other spaces with this same property include $C[0, 1]$ and $\ell_{q}^{(3)}$ for all $q > 2$.

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arXiv2020

Distance matrices of subsets of the Hamming cube

Graham and Winkler derived a formula for the determinant of the distance matrix of a full-dimensional set of $n + 1$ points $\{ x_{0}, x_{1}, \ldots , x_{n} \}$ in the Hamming cube $H_{n} = ( \{ 0,1 \}^{n}, \ell_{1} )$. In this article we derive a formula for the determinant of the distance matrix $D$ of an arbitrary set of $m + 1$ points $\{ x_{0}, x_{1}, \ldots , x_{m} \}$ in $H_{n}$. It follows from this more general formula that $\det (D) \not= 0$ if and only if the vectors $x_{0}, x_{1}, \ldots , x_{m}$ are affinely independent. Specializing to the case $m = n$ provides new insights into the original formula of Graham and Winkler. A significant difference that arises between the cases $m < n$ and $m = n$ is noted. We also show that if $D$ is the distance matrix of an unweighted tree on $n + 1$ vertices, then $\langle D^{-1} \mathbf{1}, \mathbf{1} \rangle = 2/n$ where $\mathbf{1}$ is the column vector all of whose coordinates are $1$. Finally, we derive a new proof of Murugan's classification of the subsets of $H_{n}$ that have strict $1$-negative type.

math.FA↗

arXiv2016

Comparing the generalized roundness of metric spaces

Motivated by the local theory of Banach spaces we introduce a notion of finite representability for metric spaces. This allows us to develop a new technique for comparing the generalized roundness of metric spaces. We illustrate this technique in two different ways by applying it to Banach spaces and metric trees. In the realm of Banach spaces we obtain results such as the following: (1) if $\mathcal{U}$ is any ultrafilter and $X$ is any Banach space, then the second dual $X^{\ast\ast}$ and the ultrapower $(X)_{\mathcal{U}}$ have the same generalized roundness as $X$, and (2) no Banach space of positive generalized roundness is uniformly homeomorphic to $c_{0}$ or $\ell_{p}$, $2 < p < \infty$. Our technique also leads to the identification of new classes of metric trees of generalized roundness one. In particular, we give the first examples of metric trees of generalized roundness one that have finite diameter. These results on metric trees provide a natural sequel to a paper of Caffarelli, Doust and Weston. In addition, we show that metric trees of generalized roundness one possess special Euclidean embedding properties that distinguish them from all other metric trees.

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arXiv2016

The geometry of two-valued subsets of $L_{p}$-spaces

Let $\mathcal{M}(Ω, μ)$ denote the algebra of all scalar-valued measurable functions on a measure space $(Ω, μ)$. Let $B \subset \mathcal{M}(Ω, μ)$ be a set of finitely supported measurable functions such that the essential range of each $f \in B$ is a subset of $\{ 0,1 \}$. The main result of this paper shows that for any $p \in (0, \infty)$, $B$ has strict $p$-negative type when viewed as a metric subspace of $L_{p}(Ω, μ)$ if and only if $B$ is an affinely independent subset of $\mathcal{M}(Ω, μ)$ (when $\mathcal{M}(Ω, μ)$ is considered as a real vector space). It follows that every two-valued (Schauder) basis of $L_{p}(Ω, μ)$ has strict $p$-negative type. For instance, for each $p \in (0, \infty)$, the system of Walsh functions in $L_{p}[0,1]$ is seen to have strict $p$-negative type. The techniques developed in this paper also provide a systematic way to construct, for any $p \in (2, \infty)$, subsets of $L_{p}(Ω, μ)$ that have $p$-negative type but not $q$-negative type for any $q > p$. Such sets preclude the existence of certain types of isometry into $L_{p}$-spaces.

math.FA↗

arXiv2015

The asymptotic enhanced negative type of finite ultrametric spaces

Negative type inequalities arise in the study of embedding properties of metric spaces, but they often reduce to intractable combinatorial problems. In this paper we study more quantitative versions of these inequalities involving the so-called $p$-negative type gap. In particular, we focus our attention on the class of finite ultrametric spaces which are important in areas such as phylogenetics and data mining. Let $(X,d)$ be a given finite ultrametric space with minimum non-zero distance $α$. Then the $p$-negative type gap $Γ_{X}(p)$ of $(X,d)$ is positive for all $p \geq 0$. In this paper we compute the value of the limit \begin{eqnarray*} Γ_{X}(\infty) & = & \lim\limits_{p \rightarrow \infty} \frac{Γ_{X}(p)}{α^{p}}. \end{eqnarray*} It turns out that this value is positive and it may be given explicitly by an elegant combinatorial formula. On the basis of our calculations we are then able to characterize when $Γ_{X}(p)/ α^{p}$ is constant on $[0, \infty)$. The determination of $Γ_{X}(\infty)$ also leads to new, asymptotically sharp, families of enhanced $p$-negative type inequalities for $(X,d)$. Indeed, suppose that $G \in (0, Γ_{X}(\infty))$. Then, for all sufficiently large $p$, we have \begin{eqnarray*} \frac{G \cdot α^{p}}{2} \left( \sum\limits_{k=1}^{n} |ζ_{k}| \right)^{2} + \sum\limits_{j,i =1}^{n} d(z_{j},z_{i})^{p} ζ_{j} ζ_{i} & \leq & 0 \end{eqnarray*} for each finite subset $\{ z_{1}, \ldots, z_{n} \} \subseteq X$ and each choice of real numbers $ζ_{1}, \ldots, ζ_{n}$ with $ζ_{1} + \cdots + ζ_{n} = 0$. We note that these results do not extend to general finite metric spaces.

math.MG↗

arXiv2014

Polygonal equalities and virtual degeneracy in $L_{p}$-spaces

Suppose $0 < p \leq 2$ and that $(Ω, μ)$ is a measure space for which $L_{p}(Ω, μ)$ is at least two-dimensional. The central results of this paper provide a complete description of the subsets of $L_{p}(Ω, μ)$ that have strict $p$-negative type. In order to do this we study non-trivial $p$-polygonal equalities in $L_{p}(Ω, μ)$. These are equalities that can, after appropriate rearrangement and simplification, be expressed in the form \begin{eqnarray*} \sum\limits_{j, i = 1}^{n} α_{j} α_{i} {\| z_{j} - z_{i} \|}_{p}^{p} & = & 0 \end{eqnarray*} where $\{ z_{1}, \ldots, z_{n} \}$ is a subset of $L_{p}(Ω, μ)$ and $α_{1}, \ldots, α_{n}$ are non-zero real numbers that sum to zero. We provide a complete classification of the non-trivial $p$-polygonal equalities in $L_{p}(Ω, μ)$. The cases $p < 2$ and $p = 2$ are substantially different and are treated separately. The case $p = 1$ generalizes an elegant result of Elsner, Han, Koltracht, Neumann and Zippin. Another reason for studying non-trivial $p$-polygonal equalities in $L_{p}(Ω, μ)$ is due to the fact that they preclude the existence of certain types of isometry. For example, our techniques show that if $(X,d)$ is a metric space that has strict $q$-negative type for some $q \geq p$, then: (1) $(X,d)$ is not isometric to any linear subspace $W$ of $L_{p}(Ω, μ)$ that contains a pair of disjointly supported non-zero vectors, and (2) $(X,d)$ is not isometric to any subset of $L_{p}(Ω, μ)$ that has non-empty interior. Furthermore, in the case $p = 2$, it also follows that $(X,d)$ is not isometric to any affinely dependent subset of $L_{2}(Ω, μ)$.

math.FA↗

arXiv2014

Polygonal equalities in Hilbert spaces

This work has been expanded and fully incorporated into arXiv:1203.5837. Cases of equality in the classical 2-negative type inequalities for Hilbert spaces are characterized in terms of balanced signed simplices. It follows that a metric subspace of a Hilbert space H has strict 2-negative type if and only if it is affinely independent (when H is considered as a real vector space). This allows a complete description of Shkarin's class M.

math.MG↗

arXiv2014

A direct proof that $\ell_\infty^{(3)}$ has generalized roundness zero

Metric spaces of generalized roundness zero have interesting non-embedding properties. For instance, we note that no metric space of generalized roundness zero is isometric to any metric subspace of any $L_{p}$-space for which $0 < p \leq 2$. Lennard, Tonge and Weston gave an indirect proof that $\ell_{\infty}^{(3)}$ has generalized roundness zero by appealing to highly non-trivial isometric embedding theorems of Bretagnolle Dacunha-Castelle and Krivine, and Misiewicz. In this paper we give a direct proof that $\ell_{\infty}^{(3)}$ has generalized roundness zero. This provides insight into the combinatorial geometry of $\ell_{\infty}^{(3)}$ that causes the generalized roundness inequalities to fail. We complete the paper by noting a characterization of real quasi-normed spaces of generalized roundness zero.

math.FA↗

arXiv2013

Roundness properties of ultrametric spaces

We obtain several new characterizations of ultrametric spaces in terms of roundness, generalized roundness, strict p-negative type, and p-polygonal equalities (p > 0). This allows new insight into the isometric embedding of ultrametric spaces into Euclidean spaces. We also consider roundness properties additive metric spaces which are not ultrametric.

math.FA↗

arXiv2011

Strongly non embeddable metric spaces

Enflo constructed a countable metric space that may not be uniformly embedded into any metric space of positive generalized roundness. Dranishnikov, Gong, Lafforgue and Yu modified Enflo's example to construct a locally finite metric space that may not be coarsely embedded into any Hilbert space. In this paper we meld these two examples into one simpler construction. The outcome is a locally finite metric space $(\mathfrak{Z}, ζ)$ which is strongly non embeddable in the sense that it may not be embedded uniformly or coarsely into any metric space of non zero generalized roundness. Moreover, we show that both types of embedding may be obstructed by a common recursive principle. It follows from our construction that any metric space which is Lipschitz universal for all locally finite metric spaces may not be embedded uniformly or coarsely into any metric space of non zero generalized roundness. Our construction is then adapted to show that the group $\mathbb{Z}_ω=\bigoplus_{\aleph_0}\mathbb{Z}$ admits a Cayley graph which may not be coarsely embedded into any metric space of non zero generalized roundness. Finally, for each $p \geq 0$ and each locally finite metric space $(Z,d)$, we prove the existence of a Lipschitz injection $f : Z \to \ell_{p}$.

math.FA↗

arXiv2011

Metric trees of generalized roundness one

Every finite metric tree has generalized roundness strictly greater than one. On the other hand, some countable metric trees have generalized roundness precisely one. The purpose of this paper is to identify some large classes of countable metric trees that have generalized roundness precisely one. At the outset we consider spherically symmetric trees endowed with the usual combinatorial metric (SSTs). Using a simple geometric argument we show how to determine decent upper bounds on the generalized roundness of finite SSTs that depend only on the downward degree sequence of the tree in question. By considering limits it follows that if the downward degree sequence $(d_{0}, d_{1}, d_{2}...)$ of a SST $(T,ρ)$ satisfies $|\{j \, | \, d_{j} > 1 \}| = \aleph_{0}$, then $(T,ρ)$ has generalized roundness one. Included among the trees that satisfy this condition are all complete $n$-ary trees of depth $\infty$ ($n \geq 2$), all $k$-regular trees ($k \geq 3$) and inductive limits of Cantor trees. The remainder of the paper deals with two classes of countable metric trees of generalized roundness one whose members are not, in general, spherically symmetric. The first such class of trees are merely required to spread out at a sufficient rate (with a restriction on the number of leaves) and the second such class of trees resemble infinite combs.

math.FA↗

arXiv2009

Strict p-negative type of a metric space

Doust and Weston introduced a new method called "enhanced negative type" for calculating a non trivial lower bound p(T) on the supremal strict p-negative type of any given finite metric tree (T,d). In the context of finite metric trees any such lower bound p(T) > 1 is deemed to be non trivial. In this paper we refine the technique of enhanced negative type and show how it may be applied more generally to any finite metric space (X,d) that is known to have strict p-negative type for some non negative p. This allows us to significantly improve the lower bounds on the supremal strict p-negative type of finite metric trees that were given by Doust and Weston and, moreover, leads in to one of our main results: The supremal p-negative type of a finite metric space cannot be strict. By way of application we are then able to exhibit large classes of finite 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 a metric space (finite or otherwise) has p-negative type for some p > 0, then it must have strict q-negative type for all q in [0,p). This generalizes a well known theorem of Schoenberg and leads to a complete classification of the intervals on which a metric space may have strict p-negative type. (Several of the results in this paper hold more generally for semi-metric spaces.)

math.FA↗

arXiv2008

Optimal lower bounds on the maximal p-negative type of finite metric spaces

This article derives lower bounds on the supremal (strict) p-negative type of finite metric spaces using purely elementary techniques. The bounds depend only on the cardinality and the (scaled) diameter of the underlying finite metric space. Examples show that these lower bounds can easily be best possible under clearly delineated circumstances. We further point out that the entire theory holds (more generally) for finite semi-metric spaces without modification and wherein the lower bounds are always optimal.

math.FA↗

arXiv2008

Enhanced negative type for finite metric trees

Finite metric trees are known to have strict 1-negative type. In this paper we introduce a new family of inequalities that quantify the extent of the "strictness" of the 1-negative type inequalities for finite metric trees. These inequalities of "enhanced 1-negative type" are sufficiently strong to imply that any given finite metric tree must have strict p-negative type for all values of p in an open interval that contains the number 1. Moreover, these open intervals can be characterized purely in terms of the unordered distribution of edge weights that determine the path metric on the particular tree, and are therefore largely independent of the tree's internal geometry. From these calculations we are able to extract a new non linear technique for improving lower bounds on the maximal p-negative type of certain finite metric spaces. Some pathological examples are also considered in order to stress certain technical points.

math.FA↗