arXiv subjects

Nicolò Zava

Publications and source records attributed to Nicolò Zava.

10 recordsLinked to original sources

arXiv2025

Lower Bounding the Gromov--Hausdorff distance in Metric Graphs

Let $G$ be a finite, connected metric graph and let $X\subseteq G$ be a subset. If $X$ is sufficiently dense in $G$, we show that the Gromov--Hausdorff distance matches the Hausdorff distance, namely $d_\gh(G,X)=d_\h(G,X)$. When the metric graph is the circle $G=S^1$ with circumference $2π$, a recent study established the equality $d_\gh(S^1,X)=d_\h(S^1,X)$ whenever $d_\gh(S^1,X)<\fracπ{6}$. Our results relax this hypothesis to $d_\gh(S^1,X)<\fracπ{3}$, and furthermore, we show that the constant $\fracπ{3}$ is the best possible. We lower bound the Gromov--Hausdorff distance $d_\gh(G,X)$ by the Hausdorff distance $d_\h(G,X)$ via a simple topological obstruction: the existence of a possibly discontinuous function $f\colon G \to X$ with too small distortion contradicts the connectedness of $G$.

math.MG↗

arXiv2025

Gromov-Hausdorff distance between chromatic metric pairs and stability of the six-pack

Chromatic metric pairs consist of a metric space and a coloring function partitioning a subset thereof into various colors. It is a natural extension of the notion of chromatic point sets studied in chromatic topological data analysis. A useful tool in the field is the six-pack, a collection of six persistence diagrams, summarizing homological information about how the colored subsets interact. We introduce a suitable generalization of the Gromov-Hausdorff distance to compare chromatic metric pairs. We show some basic properties and validate this definition by obtaining the stability of the six-pack with respect to that distance. We conclude by discussing its restriction to metric pairs and its role in the stability of the Čech persistence diagrams.

math.MG↗

arXiv2025

Topological dimension of the Gromov-Hausdorff and Gromov-Prokhorov spaces

The Gromov-Hausdorff distance is a dissimilarity metric capturing how far two spaces are from being isometric. The Gromov-Prokhorov distance is a similar notion for metric measure spaces. In this paper, we study the topological dimension of the Gromov-Hausdorff and Gromov-Prokhorov spaces. We show that the dimension of the space of isometry classes of metric spaces with at most $n$ points endowed with the Gromov-Hausdorff distance is $\frac{n(n-1)}{2}$, and that of mm-isomorphism classes of metric measure spaces whose support consists of $n$ points is $\frac{(n+2)(n-1)}{2}$. Hence, the spaces of all isometry classes of finite metric spaces and of all mm-isomorphism classes of finite metric measure spaces are strongly countable dimensional. If, instead, the cardinalities are not limited, the spaces are strongly infinite-dimensional.

math.MG↗

arXiv2024

Coarse and bi-Lipschitz embeddability of subspaces of the Gromov-Hausdorff space into Hilbert spaces

In this paper, we discuss the embeddability of subspaces of the Gromov-Hausdorff space, which consists of isometry classes of compact metric spaces endowed with the Gromov-Hausdorff distance, into Hilbert spaces. These embeddings are particularly valuable for applications to topological data analysis. We prove that its subspace consisting of metric spaces with at most n points has asymptotic dimension $n(n-1)/2$. Thus, there exists a coarse embedding of that space into a Hilbert space. On the contrary, if the number of points is not bounded, then the subspace cannot be coarsely embedded into any uniformly convex Banach space and so, in particular, into any Hilbert space. Furthermore, we prove that, even if we restrict to finite metric spaces whose diameter is bounded by some constant, the subspace still cannot be bi-Lipschitz embedded into any finite-dimensional Hilbert space. We obtain both non-embeddability results by finding obstructions to coarse and bi-Lipschitz embeddings in families of isometry classes of finite subsets of the real line endowed with the Euclidean-Hausdorff distance.

math.MG↗

arXiv2024

Coarse structures on locally compact abelian groups

Motivated by the study of the large-scale geometry of topological groups, we investigate particular families of subsets of topological groups named group ideals. We compare different group ideals in the realm of locally compact groups. In particular, we show that a subset of a locally compact abelian group is relatively compact if and only if it is coarsely bounded. Using this result, we prove that an infinite-dimensional Banach space cannot be embedded into any product of locally compact groups.

math.MG↗

arXiv2023

On the metric spaces of lattices and periodic point sets

Lattices and periodic point sets are well known objects from discrete geometry. They are also used in crystallography as one of the models of atomic structure of periodic crystals. In this paper we study the embedding properties of spaces of lattices and periodic point sets equipped with optimal bijection metrics (i.e., bottleneck and Euclidean bottleneck metrics). We focus our treatment on embeddings into Hilbert space. On one hand this is motivated by modern data analysis, which is mostly based on statistical approaches developed on Euclidean on Hilbert spaces, hence such embeddings play a major role in applied pipelines. On the other hand there is a well-established methodology related to such questions in coarse geometry, arising from the work on the Novikov conjecture. The main results of this paper provide different conditions, under which the spaces of lattices or periodic point sets are Lipschitz or coarsely (non)embeddable into Hilbert space. The various conditions are phrased in terms of density, packing radius, covering radius, the cardinality of the motif, and the diameter of the unit cell.

math.MG↗

arXiv2022

Weakly weighted generalised quasi-metric spaces and semilattices

Motivated by recent applications to entropy theory in dynamical systems, we generalise notions introduced by Matthews and define weakly weighted and componentwisely weakly weighted (generalised) quasi-metrics. We then systematise and extend to full generality the correspondences between these objects and other structures arising in theoretical computer science and dynamics. In particular, we study the correspondences with weak partial metrics, and, if the underlying space is a semilattice, with invariant (generalised) quasi-metrics satisfying the descending path condition, and with strictly monotone semi(-co-)valuations. We conclude discussing, for endomorphisms of generalised quasi-metric semilattices, a generalisation of both the known intrinsic semilattice entropy and the semigroup entropy.

cs.IT↗

arXiv2022

Stability of the q-hyperconvex hull of a quasi-metric space

In this paper, we study the stability of the q-hyperconvex hull of a quasi-metric space, adapting known results for the hyperconvex hull of a metric space. To pursue this goal, we extend well-known metric notions, such as Gromov-Hausdorff distance and rough isometries, to the realm of quasi-metric spaces. In particular, we prove that two q-hyperconvex hulls are close with respect to the Gromov-Hausdorff distance if so are the original spaces. Moreover, we provide an intrinsic characterisation of those spaces that are Sym-large in their q-hyperconvex hulls.

math.MG↗

arXiv2019

Categories of coarse groups: quasi-homomorphisms and functorial coarse structures

Coarse geometry is the study of large-scale properties of spaces. In this paper we study group coarse structures (i.e., coarse structures on groups that agree with the algebraic structures), by using group ideals. We introduce a large class of examples of group coarse structures induced by cardinal invariants. In order to enhance the categorical treatment of the subject, we use quasi-homomorphisms, as a large-scale counterpart of homomorphisms. In particular, the localisation of a category plays a fundamental role. We then define the notion of functorial coarse structures and we give various examples of those structures.

math.GN↗

arXiv2018

Generalisations of coarse spaces

Coarse geometry, the branch of topology that studies the global properties of spaces, was originally developed for metric spaces and then Roe introduced coarse structures as a large-scale counterpart of uniformities. In the literature, there are very important generalisations of uniform spaces, such as semi-uniform and quasi-uniform spaces. In this paper, we introduce and start to study their large-scale counterparts, which generalise coarse spaces: semi-coarse spaces and quasi-coarse spaces.

math.GN↗