In this paper we establish existence and uniqueness results for conical geodesic bicombings on subsets of normed vector spaces. Concerning existence, we give a first example of a convex geodesic bicombing that is not consistent. Furthermore, we show that under a mild geometric assumption on the norm a conical geodesic bicombing on an open subset of a normed vector space locally consists of linear geodesics. As an application, we obtain by the use of a Cartan-Hadamard type result that if a closed convex subset C of a Banach space has non-empty interior, then it admits a unique consistent conical geodesic bicombing, namely the one given by the linear segments.Recently, classical results from the theory of CAT(0) spaces have been transferred to metric spaces that admit conical geodesic bicombings, cf. [Bas17, Des16, DL16, Mie16] and [Kel16]. In the past century, notions related to conical geodesic bicombings have also been considered in metric fixed point theory, most notable W-convexity mappings, cf. [Tak70], and hyperbolic spaces in the sense of S. Reich and I. Shafrir, cf. [RS90]. It is worth to point out that the study of metric spaces that admit conical geodesic bicombing may also lead to new results about word hyperbolic groups, as every word hyperbolic group acts geometrically on a proper, finite dimensional metric space with a unique consistent conical geodesic bicombing (the definitions are given below), cf.[DL15]. The main results of this article show that the several definitions from [DL15] lead to different classes.Our first result deals with convex geodesic bicombings. From now on, we abbreviate D(X) := X × X × [0, 1]. A geodesic bicombing σ : D(X) → X is convex if the map t → d(σ pq (t), σ p q (t)) is convex on [0, 1] for all points p, q, p , q in X. Note that if the underlying metric space is not uniquely geodesic, then a conical geodesic bicombing is not necessarily convex. Examples of conical geodesic bicombings that are not convex are ubiquitous; for instance, non-convex conical geodesic bicombings may be obtained via 1-Lipschitz retractions of linear geodesics, see [DL15, Example 2.2] or Lemma 3.1. In [DL15], it is shown that metric spaces of finite combinatorial dimension in the sense of Dress, cf. [Dre84], possess at most one convex geodesic bicombing. If it exists, this unique convex geodesic bicombing, say σ : D(X) → X, has the property that it is consistent, that is, we have for all points p, q in X that im(σ p q ) ⊂ im(σ pq ) whenever p = σ pq (s) and q = σ pq (t) with 0 ≤ s ≤ t ≤ 1. Clearly, every consistent conical geodesic bicombing is convex. In Section 2, we show that the converse does not hold by proving the subsequent theorem.Theorem 1.1. There is a compact metric space that admits a convex geodesic bicombing which is not consistent.Although there is a non-consistent convex geodesic bicombing on the space considered in Section 2, this space also admits a consistent convex geodesic bicombing. We suspect that this is a general phenomenon. Question 1.2. Let (X, d) be a proper metric spac...
Local-to-global principles are spread all-around in mathematics. The classical Cartan-Hadamard Theorem from Riemannian geometry was generalized by W. Ballmann for metric spaces with non-positive curvature, and by S. Alexander and R. Bishop for locally convex metric spaces.In this paper, we prove the Cartan-Hadamard Theorem in a more general setting, namely for spaces which are not uniquely geodesic but locally possess a suitable selection of geodesics, a so-called convex geodesic bicombing.Furthermore, we deduce a local-to-global theorem for injective (or hyperconvex) metric spaces, saying that under certain conditions a complete, simplyconnected, locally injective metric space is injective. A related result for absolute 1-Lipschitz retracts follows.
Abstract: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.
Abstract. We give a necessary and sufficient condition for gluings of hyperconvex metric spaces along weakly externally hyperconvex subsets in order that the resulting space be hyperconvex. This leads to a full characterization of gluings of two isometric copies of the same hyperconvex space.Furthermore, we investigate the case of gluings of finite dimensional hyperconvex linear spaces along linear subspaces. For this purpose, we characterize the convex polyhedra in l n ∞ which are weakly externally hyperconvex.
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