Journal: Analysis and Geometry in Metric Spaces

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Publisher

De Gruyter

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ISSN

2299-3274

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2015 - 201962020 - 20242
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Publications 1 - 8 of 8
  • (In)dependence of the axioms of Λ-trees
    Item type: Journal Article
    Appenzeller, Raphael (2024)
    Analysis and Geometry in Metric Spaces
    A Λ-tree is a Λ-metric space satisfying three axioms (1), (2), and (3). We give a characterization of those ordered abelian groups Λ for which axioms (1) and (2) imply axiom (3). As a special case, it follows that for the important class of ordered abelian groups Λ that satisfy Λ = 2Λ, (3) follows from (1) and (2). For some ordered abelian groups Λ, we show that axiom (2) is independent of axioms (1) and (3) and ask whether this holds for all ordered abelian groups. Part of this work has been formalized in the proof assistant Lean.
  • The Poincaré Inequality Does Not Improve with Blow-Up
    Item type: Journal Article
    Schioppa, Andrea (2016)
    Analysis and Geometry in Metric Spaces
    For each β > 1 we construct a family Fβ of metric measure spaces which is closed under the operation of taking weak-tangents (i.e. blow-ups), and such that each element of Fβ admits a (1, P)-Poincaré inequality if and only if P > β.
  • Lipschitz Extensions to Finitely Many Points
    Item type: Journal Article
    Basso, Giuliano (2018)
    Analysis and Geometry in Metric Spaces
    We consider Lipschitz maps with values in quasi-metric spaces and extend such maps to finitely many points. We prove that in this context every 1-Lipschitz map admits an extension such that its Lipschitz constant is bounded from above by the number of added points plus one. Moreover, we prove that if the source space is a Hilbert space and the target space is a Banach space, then there exists an extension such that its Lipschitz constant is bounded from above by the square root of the total of added points plus one. We discuss applications to metric transforms.
  • Poincaré Inequalities for Mutually SingularMeasures
    Item type: Journal Article
    Schioppa, Andrea (2015)
    Analysis and Geometry in Metric Spaces
    Using an inverse system of metric graphs as in [3], we provide a simple example of a metric space X that admits Poincaré inequalities for a continuum of mutually singular measures.
  • Gluing Hyperconvex Metric Spaces
    Item type: Journal Article
    Miesch, Benjamin (2015)
    Analysis and Geometry in Metric Spaces
    We investigate how to glue hyperconvex (or injective) metric spaces such that the resulting space remains hyperconvex. We give two new criteria, saying that on the one hand gluing along strongly convex subsets and on the other hand gluing along externally hyperconvex subsets leads to hyperconvex spaces. Furthermore, we show by an example that these two cases where gluing works are opposed and cannot be combined.
  • Flats in Spaces with Convex Geodesic Bicombings
    Item type: Journal Article
    Descombes, Dominic; Lang, Urs (2016)
    Analysis and Geometry in Metric Spaces
    In spaces of nonpositive curvature the existence of isometrically embedded at (hyper)planes is often granted by apparently weaker conditions on large scales.We show that some such results remain valid for metric spaces with non-unique geodesic segments under suitable convexity assumptions on the distance function along distinguished geodesics. The discussion includes, among other things, the Flat Torus Theorem and Gromov’s hyperbolicity criterion referring to embedded planes. This generalizes results of Bowditch for Busemann spaces.
  • In nitesimal Structure of Di erentiability Spaces, and Metric Di erentiation
    Item type: Journal Article
    Cheeger, Jeff; Kleiner, Bruce; Schioppa, Andrea (2016)
    Analysis and Geometry in Metric Spaces
    We prove metric differentiation for differentiability spaces in the sense of Cheeger [10, 14, 27]. As corollaries we give a new proof of one of the main results of [14], a proof that the Lip-lip constant of any Lip-lip space in the sense of Keith [27] is equal to 1, and new nonembeddability results.
  • Lipschitz Chain Approximation of Metric Integral Currents
    Item type: Journal Article
    Goldhirsch, Tommaso (2022)
    Analysis and Geometry in Metric Spaces
    Every integral current in a locally compact metric space X can be approximated by a Lipschitz chain with respect to the normal mass, provided that Lipschitz maps into X can be extended slightly.
Publications 1 - 8 of 8