DOCSLIB.ORG

Geometry and Structure of Metric Injective Hulls

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Geometry and Structure of Metric Injective Hulls Research Collection Doctoral Thesis Geometry and structure of metric injective hulls Author(s): Pavón, Maël Publication Date: 2016 Permanent Link: https://doi.org/10.3929/ethz-a-010633534 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library Diss. ETH No. 23120 Geometry and Structure of Metric Injective Hulls A thesis submitted to attain the degree of DOCTOR OF SCIENCES OF ETH ZURICH (Dr. sc. ETH Zurich) presented by Mael¨ Pavon´ MSc ETH Math., ETH Zurich born September 23rd, 1989 citizen of Haut-Intyamon (FR) and Spain accepted on the recommendation of Prof. Dr. Urs Lang, Examiner Prof. Dr. Victor Chepoi, Co-Examiner 2016 Abstract In this thesis, we prove various results on metric injective hulls and injective metric spaces. In Chapter II, we show that the operator given by assigning to a metric space the corresponding metric space defined by Isbell's injective hull is 2-Lipschitz in the Gromov-Hausdorff sense when defined on the class of general metric spaces and 1-Lipschitz when restricted to the class of tree-like metric spaces. These estimates are optimal. In Chapter III, we character- ize effectively the injective affine subspaces of the finite dimensional injective normed spaces in any dimension and go on characterizing effectively injective convex polyhedra, this characterization provides in particular a concrete veri- fication algorithm. We then make use of this result to prove that the solution set of a system of linear inequalities with at most two variables per inequality is injective if it is non-empty. Turning to injective hulls with the structure of a polyhedral complex, we extend in Chapter IV the canonical decomposi- tion theory of Bandelt and Dress to infinite metric spaces with integer-valued metric. We consider infinite totally split-decomposable metric spaces with integer-valued metric and satisfying a local rank condition. We then give a characterization for Isbell's injective hull of such metric spaces to be combi- natorially equivalent to a cube complex satisfying the CAT(0) link condition. We apply this, among others, to injective hulls of cycle graphs. In Chapter V, we give an alternative characterization of finite combinatorial dimension for metric spaces. We consider the canonical decomposition of the collection of extremal functions that induce admissible graphs. We prove an optimal bound on the diameters of the elements of this canonical decomposition for discretely path-connected metric spaces. We conclude by proving for different classes of metric spaces including any proper metric space, that such a space is injective if and only if it is 4-hyperconvex and possesses a geodesic bicombing. ii Zusammenfassung In dieser Dissertation beweisen wir Resultate ¨uber metrische injektive H¨ullen und injektive metrische R¨aume. In Kapitel II beweisen wir, dass die Is- bellsche injektive H¨ulleein 2-Lipschitz-Operator ist im Sinne von Gromov- Hausdorff, wenn man sie auf der Klasse aller metrischen R¨aumedefiniert. Die Isbellsche injektive H¨ullewird zu einem 1-Lipschitz-Operator, wenn man sie auf die Klasse aller baumartigen metrischen R¨aumeeinschr¨ankt. In Kapitel III charakterisieren wir effektiv die injektiven affinen Teilmengen der endlich dimensionalen normierten R¨aumebeliebiger Dimension. Ausserdem beweisen wir noch eine effektive Charakterisierung aller injektiven konvexen Polyeder. Diese Charakterisierung liefert insbesondere einen konkreten Verifikationsal- gorithmus. Wir verwenden dieses Resultat, um zu beweisen, dass ein System linearer Ungleichungen mit h¨ochstens zwei Variablen pro Ungleichung injek- tiv ist, wenn es nicht leer ist. Desweiteren betrachten wir diejenigen injek- tiven H¨ullen,die die Struktur eines polyedrischen Komplexes haben und in Kapitel IV verallgemeinern wir die kanonische Zerlegungstheorie von Bandelt und Dress, indem wir sie f¨ur unendliche metrische R¨aumemit ganzzahliger Metrik entwickeln. Wir betrachten dann diejenigen unendlichen, vollst¨andig split-zerlegbaren metrischen R¨aumenmit ganzzahliger Metrik, die eine lokale Rangbedingung erf¨ullen.F¨ur diese Klasse metrischer R¨aumecharakterisieren wir diejenigen, f¨urdie ihre Isbellsche injektive H¨ullekombinatorisch ¨aquivalent zu einem W¨urfelkomplex ist, welcher die CAT(0)-Link-Bedingung erf¨ullt.Wir wenden dieses Kriterium insbesondere auf Zykelgraphen an. In Kapitel V geben wir eine alternative Charakterisierung der endlichen kombinatorischen Dimension. Wir betrachten die kanonische Zerlegung der Familie aller Ex- tremalfunktionen, die zul¨assige Graphen induzieren. Wir beweisen eine opti- male Schranke an die Durchmesser der Elemente dieser kanonischen Zerlegung f¨urdiskret wegzusammenh¨angendemetrische R¨aume. Schliesslich beweisen wir f¨urverschiedene Klassen metrischer R¨aume,insbesondere f¨urjeden soge- nannten proper metrischen Raum, dass dieser genau dann injektiv ist, wenn er 4-hyperkonvex ist und ein Bicombing besitzt. iii Acknowledgements I would like to thank my advisor Prof. Urs Lang for his guidance and for the time he dedicated to me all along my PhD studies. I would like to thank Prof. Victor Chepoi for being my Co-Referee. I gratefully acknowledge the support of the Swiss National Science Foundation. Finally, I would like to thank all the members of the research team of Prof. Urs Lang at ETH Z¨urich. iv Contents I Introduction 1 II Metric Stability of Trees and Tight Spans 8 II.1 Introduction . 8 II.2 Extension of Roughly Isometric Relations . 10 II.3 Gromov–Hausdorff Distance Estimates . 14 III Injective Convex Polyhedra 18 III.1 Introduction . 18 n III.2 Injective Linear Subspaces in l1 . 22 III.3 Tangent cones . 24 III.4 Systems of Inequalities . 27 III.5 The Cone KC ........................... 30 III.6 Injective Convex Polyhedral Cones . 38 III.7 Graph Representation of Systems . 44 III.8 Injectivity of Systems . 47 IV Injective Hulls of Infinite Totally Split-Decomposable Metric Spaces 52 IV.1 Introduction . 52 IV.2 Decomposition Theory . 58 IV.3 The Buneman Complex . 75 IV.4 The CAT(0) Link Condition . 100 IV.5 Examples . 108 V Further Results on Metric Injectivity 119 V.1 Introduction . 119 V.2 Dress' Theorem . 124 V.3 Barriers, Graphs of Maximal Elements and Applications . 132 V.4 The Diameter of Cells of E(X) . 139 v CONTENTS V.5 Bicombings . 145 vi Chapter I Introduction The present thesis consists of four main components, corresponding to works [31, 37, 38] together with a collection of further results, each one of these components focusing on a precise topic of the theory of metric spaces. The common goal of these four works is to develop tools for the study of injective hulls and injective metric spaces in view of many different applications. For instance, in phylogenetic analysis where injective hulls are used to compare general metrics to tree-like ones, cf. [19, 20]. Furthermore, injective hulls appear to be relevant in theoretical computer science in particular for online algorithms in relation with the k-server problem, cf. [13, 14]. Additionally, injective hulls provide a source of new techniques and approaches for purely mathematical questions with geometric elements. As an illustration, injectivity can be regarded as a general metric notion of global weak non-positive curvature since injective metric spaces share common features with CAT(0) spaces like, for instance, the existence of a geodesic bicombing. Developing tools for injective metric spaces and hulls can therefore lead for example to sharpen and understand better known results of CAT(0) geometry. A metric space (X; d) is called injective if for any isometric embedding i : A ! B of metric spaces and any 1-Lipschitz (equivalently distance- nonincreasing) map f : A ! X, there exists a 1-Lipschitz map g : B ! X, so that g ◦ i = f. Examples of such spaces include the real line R, l1(I) for any index set I, and all complete metric trees. As can be deduced from the definition, injective metric spaces are in particular non-empty, complete, geodesic and every triple of points has at least one median point. Moreover, injective metric spaces are abolute 1-Lipschitz retracts and reciprocally ev- ery absolute 1-Lipschitz retract is injective. In addition, all injective metric 1 I. INTRODUCTION spaces are contractible. The terminology coincides with the definition of an injective object in category theory. Accordingly and unless otherwise stated, by injective we mean in the category of metric spaces and 1-Lipschitz maps. As a matter of fact, proving injectivity is often achieved indirectly via an equivalent but more handy criterion. One such equivalent criterion, which essentially amounts to a point by point extension property, is hyperconvexity, which was introduced in [2]. A metric space (X; d) is called hyperconvex if any collection f(xi; ri)gi2I ⊂ X × [0; 1) with the property that d(xi; xj) ≤ ri + rj T for all pair of indices i; j 2 I, satisfies i2I B(xi; ri) 6= ; where B(x; r) := fy 2 X : d(x; y) ≤ rg. With the help of Zorn's lemma, it is not difficult to prove equivalence between injectivity and hyperconvexity. By the hyperconvexity criterion together with a theorem of Nachbin proved in [36], it follows that a normed space is linearly injective (i.e., injective in the category of normed spaces and linear 1-Lipschitz maps) if and only if it is injective in the metric category. Ubiquity of injectivity in the theory of metric spaces is best appreciated when considering Isbell's injective hull construction. Indeed, Isbell showed in [29], that every metric space (X; d) possesses an injective hull (e; E(X)), by which is meant that two properties hold, namely that E(X) is an injective metric space (endowed with a canonical metric which is for conciseness absent from the notation) and e: X ! E(X) is an isometric embedding such that every isometric embedding of X into another injective metric space factors through e.
Load more

Recommended publications
  • Metric Spaces in Pure and Applied Mathematics
    Documenta Math. 121 Metric Spaces in Pure and Applied Mathematics A. Dress, K. T. Huber1, V. Moulton2 Received: September 25, 2001 Revised: November 5, 2001 Communicated by Ulf Rehmann Abstract. The close relationship between the theory of quadratic forms and distance analysis has been known for centuries, and the the- ory of metric spaces that formalizes distance analysis and was devel- oped over the last century, has obvious strong relations to quadratic- form theory. In contrast, the first paper that studied metric spaces as such – without trying to study their embeddability into any one of the standard metric spaces nor looking at them as mere ‘presentations’ of the underlying topological space – was, to our knowledge, written in the late sixties by John Isbell. In particular, Isbell showed that in the category whose objects are metric spaces and whose morphisms are non-expansive maps, a unique injective hull exists for every object, he provided an explicit construction of this hull, and he noted that, at least for finite spaces, it comes endowed with an intrinsic polytopal cell structure. In this paper, we discuss Isbell’s construction, we summarize the his- tory of — and some basic questions studied in — phylogenetic analysis, and we explain why and how these two topics are related to each other. Finally, we just mention in passing some intriguing analogies between, on the one hand, a certain stratification of the cone of all metrics de- fined on a finite set X that is based on the combinatorial properties of the polytopal cell structure of Isbell’s injective hulls and, on the other, various stratifications of the cone of positive semi-definite quadratic forms defined on Rn that were introduced by the Russian school in the context of reduction theory.
    [Show full text]
  • Helly Groups
    HELLY GROUPS JER´ EMIE´ CHALOPIN, VICTOR CHEPOI, ANTHONY GENEVOIS, HIROSHI HIRAI, AND DAMIAN OSAJDA Abstract. Helly graphs are graphs in which every family of pairwise intersecting balls has a non-empty intersection. This is a classical and widely studied class of graphs. In this article we focus on groups acting geometrically on Helly graphs { Helly groups. We provide numerous examples of such groups: all (Gromov) hyperbolic, CAT(0) cubical, finitely presented graph- ical C(4)−T(4) small cancellation groups, and type-preserving uniform lattices in Euclidean buildings of type Cn are Helly; free products of Helly groups with amalgamation over finite subgroups, graph products of Helly groups, some diagram products of Helly groups, some right- angled graphs of Helly groups, and quotients of Helly groups by finite normal subgroups are Helly. We show many properties of Helly groups: biautomaticity, existence of finite dimensional models for classifying spaces for proper actions, contractibility of asymptotic cones, existence of EZ-boundaries, satisfiability of the Farrell-Jones conjecture and of the coarse Baum-Connes conjecture. This leads to new results for some classical families of groups (e.g. for FC-type Artin groups) and to a unified approach to results obtained earlier. Contents 1. Introduction 2 1.1. Motivations and main results 2 1.2. Discussion of consequences of main results 5 1.3. Organization of the article and further results 6 2. Preliminaries 7 2.1. Graphs 7 2.2. Complexes 10 2.3. CAT(0) spaces and Gromov hyperbolicity 11 2.4. Group actions 12 2.5. Hypergraphs (set families) 12 2.6.
    [Show full text]
  • Injective Hulls of Certain Discrete Metric Spaces and Groups
    Injective hulls of certain discrete metric spaces and groups Urs Lang∗ July 29, 2011; revised, June 28, 2012 Abstract Injective metric spaces, or absolute 1-Lipschitz retracts, share a number of properties with CAT(0) spaces. In the 1960es, J. R. Isbell showed that every metric space X has an injective hull E(X). Here it is proved that if X is the vertex set of a connected locally finite graph with a uniform stability property of intervals, then E(X) is a locally finite polyhedral complex with n finitely many isometry types of n-cells, isometric to polytopes in l∞, for each n. This applies to a class of finitely generated groups Γ, including all word hyperbolic groups and abelian groups, among others. Then Γ acts properly on E(Γ) by cellular isometries, and the first barycentric subdivision of E(Γ) is a model for the classifying space EΓ for proper actions. If Γ is hyperbolic, E(Γ) is finite dimensional and the action is cocompact. In particular, every hyperbolic group acts properly and cocompactly on a space of non-positive curvature in a weak (but non-coarse) sense. 1 Introduction A metric space Y is called injective if for every metric space B and every 1- Lipschitz map f : A → Y defined on a set A ⊂ B there exists a 1-Lipschitz extension f : B → Y of f. The terminology is in accordance with the notion of an injective object in category theory. Basic examples of injective metric spaces are the real line, all complete R-trees, and l∞(I) for an arbitrary index set I.
    [Show full text]
  • Helly Meets Garside and Artin
    Invent. math. (2021) 225:395–426 https://doi.org/10.1007/s00222-021-01030-8 Helly meets Garside and Artin Jingyin Huang1 · Damian Osajda2,3 Received: 7 June 2019 / Accepted: 7 January 2021 / Published online: 15 February 2021 © The Author(s) 2021 Abstract A graph is Helly if every family of pairwise intersecting combina- torial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, such groups act geometrically on spaces with a convex geodesic bicombing, equipping them with a nonpositive-curvature-like structure. That structure has many properties of a CAT(0) structure and, addi- tionally, it has a combinatorial flavor implying biautomaticity. As immediate consequences we obtain new results for FC-type Artin groups (in particular braid groups and spherical Artin groups) and weak Garside groups, including e.g. fundamental groups of the complements of complexified finite simplicial arrangements of hyperplanes, braid groups of well-generated complex reflec- tion groups, and one-relator groups with non-trivial center. Among the results are: biautomaticity, existence of EZ and Tits boundaries, the Farrell–Jones conjecture, the coarse Baum–Connes conjecture, and a description of higher B Damian Osajda [email protected] Jingyin Huang [email protected] 1 Department of Mathematics, The Ohio State University, 100 Math Tower, 231 W 18th Ave, Columbus, OH 43210, USA 2 Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50–384 Wrocław, Poland 3 Institute of Mathematics, Polish Academy of Sciences, Sniadeckich´ 8, 00-656 Warsaw, Poland 123 396 J.
    [Show full text]
  • Spaces with Convex Geodesic Bicombings
    Research Collection Doctoral Thesis Spaces with convex geodesic bicombings Author(s): Descombes, Dominic Publication Date: 2015 Permanent Link: https://doi.org/10.3929/ethz-a-010584573 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library Diss. ETH No. 23109 Spaces with Convex Geodesic Bicombings A thesis submitted to attain the degree of Doctor of Sciences of ETH Zurich presented by Dominic Descombes Master of Science ETH in Mathematics citizen of Lignières, NE and citizen of Italy accepted on the recommendation of Prof. Dr. Urs Lang, examiner Prof. Dr. Alexander Lytchak, co-examiner 2015 Life; full of loneliness, and misery, and suffering, and unhappiness — and it's all over much too quickly. – Woody Allen Abstract In the geometry of CAT(0) or Busemann spaces every pair of geodesics, call them α and β, have convex distance; meaning d ◦ (α, β) is a convex function I → R provided the geodesics are parametrized proportional to arc length on the same interval I ⊂ R. Therefore, geodesics ought to be unique and thus even many normed spaces do not belong to these classes. We investigate spaces with non-unique geodesics where there exists a suitable selection of geodesics exposing the said (or a similar) convexity property; this structure will be called a bicombing. A rich class of such spaces arises naturally through the construction of the injective hull for arbitrary metric spaces or more generally as 1- Lipschitz retracts of normed spaces.
    [Show full text]
  • T-Theory and Analysis of Online Algorithms
    UNLV Retrospective Theses & Dissertations 1-1-2007 T-theory and analysis of online algorithms James Oravec University of Nevada, Las Vegas Follow this and additional works at: https://digitalscholarship.unlv.edu/rtds Repository Citation Oravec, James, "T-theory and analysis of online algorithms" (2007). UNLV Retrospective Theses & Dissertations. 2182. http://dx.doi.org/10.25669/no3p-11tt This Thesis is protected by copyright and/or related rights. It has been brought to you by Digital Scholarship@UNLV with permission from the rights-holder(s). You are free to use this Thesis in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Thesis has been accepted for inclusion in UNLV Retrospective Theses & Dissertations by an authorized administrator of Digital Scholarship@UNLV. For more information, please contact [email protected]. T-THEORY AND ANALYSIS OF ONLINE ALGORITHMS by James Oravec Bachelor of Arts in Computer Science University of Nevada, Las Vegas 2W05 Master of Science in Computer Science University of Nevada, Las Vegas 2007 A thesis submitted in partial fulfillment of the requirements for the Master of Science in Computer Science Department of Computer Science Howard R. Hughes College of Engineering Graduate College University of Nevada, Las Vegas August 2007 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 1448414 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted.
    [Show full text]
  • Hyperconvexity and Tight Span Theory for Diversities
    Hyperconvexity and Tight-Span Theory for Diversities David Bryanta,∗, Paul F. Tupperb aDept. of Mathematics and Statistics, University of Otago. PO Box 56 Dunedin 9054, New Zealand. Ph (64)34797889. Fax (64)34798427 bDept. of Mathematics, Simon Fraser University. 8888 University Drive, Burnaby, British Columbia V5A 1S6, Canada. Ph (778)7828636. Fax (778)7824947 Abstract The tight span, or injective envelope, is an elegant and useful construction that takes a metric space and returns the smallest hyperconvex space into which it can be embedded. The concept has stimulated a large body of theory and has applications to metric classification and data visualisation. Here we introduce a generalisation of metrics, called diversities, and demonstrate that the rich theory associated to metric tight spans and hyperconvexity extends to a seemingly richer theory of diversity tight spans and hyperconvexity. Keywords: Tight span; Injective hull; Hyperconvex; Diversity; Metric geometry; 1. Introduction Hyperconvex metric spaces were defined by Aronszajn and Panitchpakdi in [1] as part of a program to generalise the Hahn-Banach theorem to more general metric spaces (reviewed in [2], and below). Isbell [3] and Dress [4] showed that, for every metric space, there exists an essentially unique \minimal" hyperconvex arXiv:1006.1095v5 [math.MG] 23 Jan 2013 space into which that space could be embedded, called the tight span or injective envelope. Our aim is to show that the notion of hyperconvexity, the tight span, and much of the related theory can be extended beyond metrics to a class of multi-way metrics which we call diversities. ∗Corresponding author Email addresses: [email protected] (David Bryant), [email protected] (Paul F.
    [Show full text]
  • Arxiv:2002.06895V2 [Math.GR] 19 Feb 2021 3.3
    HELLY GROUPS JER´ EMIE´ CHALOPIN, VICTOR CHEPOI, ANTHONY GENEVOIS, HIROSHI HIRAI, AND DAMIAN OSAJDA Abstract. Helly graphs are graphs in which every family of pairwise intersecting balls has a non-empty intersection. This is a classical and widely studied class of graphs. In this article we focus on groups acting geometrically on Helly graphs { Helly groups. We provide numerous examples of such groups: all (Gromov) hyperbolic, CAT(0) cubical, finitely presented graph- ical C(4)−T(4) small cancellation groups, and type-preserving uniform lattices in Euclidean buildings of type Cn are Helly; free products of Helly groups with amalgamation over finite subgroups, graph products of Helly groups, some diagram products of Helly groups, some right- angled graphs of Helly groups, and quotients of Helly groups by finite normal subgroups are Helly. We show many properties of Helly groups: biautomaticity, existence of finite dimensional models for classifying spaces for proper actions, contractibility of asymptotic cones, existence of EZ-boundaries, satisfiability of the Farrell-Jones conjecture and of the coarse Baum-Connes conjecture. This leads to new results for some classical families of groups (e.g. for FC-type Artin groups) and to a unified approach to results obtained earlier. Contents 1. Introduction 2 1.1. Motivations and main results 2 1.2. Discussion of consequences of main results 5 1.3. Organization of the article and further results 6 2. Preliminaries 7 2.1. Graphs 7 2.2. Complexes 10 2.3. CAT(0) spaces and Gromov hyperbolicity 11 2.4. Group actions 12 2.5. Hypergraphs (set families) 12 2.6.
    [Show full text]
  • Persistence, Metric Invariants, and Simplification
    Persistence, Metric Invariants, and Simplification Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Osman Berat Okutan, M.S. Graduate Program in Mathematics The Ohio State University 2019 Dissertation Committee: Facundo M´emoli,Advisor Matthew Kahle Jean-Franc¸oisLafont c Copyright by Osman Berat Okutan 2019 Abstract Given a metric space, a natural question to ask is how to obtain a simpler and faithful approximation of it. In such a situation two main concerns arise: How to construct the approximating space and how to measure and control the faithfulness of the approximation. In this dissertation, we consider the following simplification problems: Finite approx- imations of compact metric spaces, lower cardinality approximations of filtered simplicial complexes, tree metric approximations of metric spaces and finite metric graph approxima- tions of compact geodesic spaces. In each case, we give a simplification construction, and measure the faithfulness of the process by using the metric invariants of the original space, including the Vietoris-Rips persistence barcodes. ii For Esra and Elif Beste iii Acknowledgments First and foremost I'd like to thank my advisor, Facundo M´emoli,for our discussions and for his continual support, care and understanding since I started working with him. I'd like to thank Mike Davis for his support especially on Summer 2016. I'd like to thank Dan Burghelea for all the courses and feedback I took from him. Finally, I would like to thank my wife Esra and my daughter Elif Beste for their patience and support.
    [Show full text]
  • Optimal and Hereditarily Optimal Realizations of Metric Spaces
    UPPSALA DISSERTATIONS IN MATHEMATICS 52 Optimal and Hereditarily Optimal Realizations of Metric Spaces Alice Lesser Department of Mathematics Uppsala University UPPSALA 2007 !" #""$ "%& ' ' ' ( ) * + ( ,( #""$( - . - /0 ' 1 2 ( 3- ' 45( &#( $" ( ( 627 8$9:8:&";:8;$:;( ) ' ' ' ' ' ' % * * < *( ) 4 * : =( 6 > 0 ' ' ? * * 0 ' = ' ( 6 6 * ? = :4 ' * ( ' : 4 * ? ( 6 66 * * ' ' 0 ' ' ? ( 6 666 * ' @ ' * * 0 * ' ? ( . * ' 0 * ( 6A *4 > % * * ' 0 ? B ' 0 ' * ' ' 0 ? >( 1 * *4 > ' ' * #: 0 0 * 4 7 ): ' ! "#$ % &'()*$+ & , - . ! ! )/# % &'()*0" & C , #""$ 622 D":#"D8 627 8$9:8:&";:8;$:; % %%% :9#8$ 3 %EE (4(E B F % %%% :9#8$5 “Don’t be too sure,” said the child patiently, “for one of the nicest things about mathematics, or anything else you might care to learn, is that many of the things which can never be, often are. You see,” he went on, “it’s very much like your trying to reach Infinity. You know that it’s there, but you just don’t know where — but just because you can never reach it doesn’t mean that it’s not worth looking for.” Norton Juster, The Phantom Tollbooth Cover image:
    [Show full text]
  • Vietoris-Rips Persistent Homology, Injective Metric Spaces, and the Filling Radius
    Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius Sunhyuk Lim1, Facundo Memoli´ 2, and Osman Berat Okutan3 1Department of Mathematics, The Ohio State University, [email protected] 2Department of Mathematics and Department of Computer Science and Engineering, The Ohio State University, [email protected] 3Department of Mathematics, Florida State University, [email protected] September 15, 2020 Abstract In the applied algebraic topology community, the persistent homology induced by the Vietoris-Rips simplicial filtration is a standard method for capturing topological information from metric spaces. In this paper, we consider a different, more geometric way of generating persistent homology of metric spaces which arises by first embedding a given metric space into a larger space and then considering thickenings of the original space inside this ambient metric space. In the course of doing this, we construct an appropriate category for studying this notion of persistent homology and show that, in a category theoretic sense, the standard persistent homology of the Vietoris-Rips filtration is isomorphic to our geometric persistent homology provided that the ambient metric space satisfies a property called injectivity. As an application of this isomorphism result we are able to precisely characterize the type of intervals that appear in the persistence barcodes of the Vietoris-Rips filtration of any com- pact metric space and also to give succinct proofs of the characterization of the persistent homology of products and metric gluings of metric spaces. Our results also permit proving arXiv:2001.07588v2 [math.AT] 14 Sep 2020 several bounds on the length of intervals in the Vietoris-Rips barcode by other metric invari- ants.
    [Show full text]
  • General Definitions
    Chapter 1 General Definitions 1.1 Basic definitions • Distance Let X be a set. A function d : X × X → R is called a distance (or dissimilarity)onX if, for all x, y ∈ X, there holds: 1. d(x, y) ≥ 0(non-negativity). 2. d(x, y)=d(y,x)(symmetry). 3. d(x, x)=0(reflexivity). In Topology, the distance d with d(x, y) = 0 implying x = y is called a symmetric. A distance which is a squared metric is called a quadrance. For any distance d, the function D1, defined for x = y by D1(x, y)= d(x, y)+c, where c = maxx,y,z∈X (d(x, y)−d(x, z)−d(y,z)), and D(x, x)= c 0, is a metric. Also, D2(x, y)=d(x, y) is a metric for sufficiently small c ≥ 0. The function D3(x, y)=inf i d(zi,zi+1), where the infimum is taken over all sequences x = z0,...,zn+1 = y,isasemi-metric. • Distance space A distance space (X, d)isasetX equipped with a distance d. • Similarity Let X be a set. A function s : X × X → R is called a similarity on X if s is non-negative, symmetric, and if s(x, y) ≤ s(x, x) holds for all x, y ∈ X, with equality if and only if x = y. The main transforms used to obtain a distance (dissimilarity) d√from a − 1−s − similarity s bounded by 1 from above are: d =1 s, d = s , d = 1 s, d = 2(1 − s2), d = arccos s, d = − ln s (cf.
    [Show full text]