Conical geodesic bicombings on subsets of normed vector spaces

Abstract

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...

The Cartan–Hadamard Theorem for metric spaces with local geodesic bicombings

“…As we show in a subsequent paper joined with G. Basso [5], this leads to a uniqueness result for convex geodesic bicombings on convex subsets of certain Banach spaces.…”

Extending and improving conical bicombings

“…As pointed out above, conical bicombings on an open subset of an injective Banach space are locally given by linear segments. In view of this result, the question has been raised whether a closed convex subset of a Banach space with non-empty interior only admits one conical bicombing, see [4,Question 1.6]. Motivated by the proof of Theorem 1.1, we found a counterexample to this question.…”

“…They have been employed considerably under various names in metric fixed point theory [42,32,27,26]. We refer the reader to the recent articles [10,4] for a systematic study of conical bicombings. Our main result, Theorem 1.1, implies that the class of metric spaces admitting a conical bicombing coincides with the class of σ-convex subsets of injective metric spaces.…”

“…The author and Miesch [4] showed that every complete metric space with a conical bicombing admits a reversible conical bicombing. Hence, as stated above, by Theorem 1.1, the class of metric spaces admitting a conical bicombing agrees with the class of σ-convex subsets of injective metric spaces.…”

Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer

“…The first condition on Y , generalizes the existence of a geodesic bicombing, as studied by [96,45,18,113,19], and is analogous in spirit to the idea of a simplicial topological space from homotopy theory [136,Chapter 83.2] and from fuzzy set theory [16], and the peaked partitions of By [68,Theorem 12.1] every subset of R d has the doubling property (see Section 6 for the definition and [68, Section 10.13] for details), and following the discussion on [27, page 3] a metric space is doubling if and only if its metric capacity is finite for all δ ∈ (0, 1].…”