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.