The linear projection constant Π(E) of a finite-dimensional real Banach space E is the smallest number C ∈ [0, +∞) such that E is a C-absolute retract in the category of real Banach spaces with bounded linear maps. We denote by Π n the maximal linear projection constant amongst n-dimensional Banach spaces. In this article, we prove that Π n may be determined by computing eigenvalues of certain two-graphs. From this result we obtain that the relative projection constants of codimension n converge to 1 + Π n . Furthermore, using the classification of K 4 -free two-graphs, we give an alternative proof of Π 2 = 4 3 . We also show by means of elementary functional analysis that for each integer n 1 there exists a polyhedral n-dimensional Banach space F n such that Π(F n ) = Π n . arXiv:1901.07866v1 [math.MG] 23 Jan 2019 4 3 , which Grünbaum conjectured to be the maximal value of Π(·) amongst 2-dimensional Banach spaces. In 2010, Chalmers and Lewicki presented an intricate proof of Grünbaum's conjecture employing the implicit function theorem and Lagrange multipliers, cf. [CL10].Our main result, see Theorem 1.2, provides a characterization of the number Π n in terms of certain maximal sums of eigenvalues of two-graphs that are K n+2 -free. In [FF84], Frankl and Füredi give a full description of two-graphs that are K 4 -free. Via this description and Theorem 1.2 we can derive from first principles that Π 2 = 4 3 . This is done in Section 4.
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...
We derive two fixed point theorems for a class of metric spaces that includes all Banach spaces and all complete Busemann spaces. We obtain our results by the use of a 1-Lipschitz barycenter construction and an existence result for invariant Radon probability measures. Furthermore, we construct a bounded complete Busemann space that admits an isometry without fixed points.
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.
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