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Fermat–Weber points with respect to an asymmetric tropical distance function are studied. It turns out that they correspond to the optimal solutions of a transportation problem. The results are applied to obtain a new method for computing consensus trees in phylogenetics. This method has several desirable properties; e.g., it is Pareto and co-Pareto on rooted triplets.

The main goal of this article is to show that for every (reflexive) infinite-dimensional Banach space X there exists a reflexive Banach space Y and T , R ∈ L ( X , Y ) such that R is a rank-one operator, ‖ T + R ‖ > ‖ T ‖ but T + R does not attain its norm. This answers a question posed by Dantas and the first two authors. Furthermore, motivated by the parallelism exhibited in the literature between the V -property introduced by Khatskevich, Ostrovskii and Shulman and the weak maximizing property introduced by Aron, García, Pellegrino and Teixeira, we also study the relationship between these two properties and norm-attaining perturbations of operators.

In this paper, we introduce the notion of a quasimodular and we prove that the respective Minkowski functional of the unit quasimodular ball becomes a quasinorm. In this way, we refer to and complete the well-known theory related to the notions of a modular and a convex modular that lead to the F -norm and to the norm, respectively. We use the obtained results to consider the basic properties of quasinormed Calderón–Lozanovskiĭ spaces E φ , where the lower Matuszewska–Orlicz index α φ plays the key role. Our studies are conducted in a full possible generality.

We study the best coapproximation problem in Banach spaces, by using Birkhoff–James orthogonality techniques. We introduce two special types of subspaces, christened the anti-coproximinal subspaces and the strongly anti-coproximinal subspaces. We obtain a necessary condition for the strongly anti-coproximinal subspaces in a reflexive Banach space whose dual space satisfies the Kadets–Klee Property. On the other hand, we provide a sufficient condition for the strongly anti-coproximinal subspaces in a general Banach space. We also characterize the anti-coproximinal subspaces of a smooth Banach space. Further, we study these special subspaces in a finite-dimensional polyhedral Banach space and find some interesting geometric structures associated with them.

In this paper, we study the extension problems of norm-additive maps between the positive unit spheres of ℓ q ( ℓ p ) , 1 ≤ p , q ≤ ∞ . Let S ℓ q ( ℓ p ) + = { x ∈ ℓ q ( ℓ p ) : x ≥ 0 ; ‖ x ‖ = 1 } be the positive unit sphere of ℓ q ( ℓ p ) , f : S ℓ q ( ℓ p ) + → S ℓ q ( ℓ p ) + be a bijective norm-additive map (preserving norm of sums), i.e., ‖ f ( x ) + f ( y ) ‖ = ‖ x + y ‖ , for all x , y ∈ S ℓ q ( ℓ p ) + . In the cases when 1 < p , q ≤ ∞ or 1 < p < ∞ , q = 1 , we show that f can be extended to a linear surjective isometry from ℓ q ( ℓ p ) onto itself. Counter examples for the remaining cases when p = 1 , 1 ≤ q ≤ ∞ or p = ∞ , q = 1 are also presented.

We investigate first Baire functionals on the dual ball of a separable Banach space X which are pointwise limit of a sequence of X whose closed span does not contain any copy of ℓ 1 (or has separable dual). We propose an example of a C ( K ) space where not all such first Baire functionals exhibit this behavior. As an application, we study a quantitative version, in terms of descriptive set theory, of family a separable Banach spaces with this peculiarity.

In uniformly convex Banach spaces, we study within this research Fibonacci–Ishikawa iteration for monotone asymptotically nonexpansive mappings. In addition to demonstrating strong convergence, we establish weak convergence result of the Fibonacci–Ishikawa sequence that generalizes many results in the literature. If the norm of the space is monotone, our consequent result demonstrates the convergence type to the weak limit of the sequence of minimizing sequence of a function. One of our results characterizes a family of Banach spaces that meet the weak Opial condition. Finally, using our iterative procedure, we approximate the solution of the Caputo-type nonlinear fractional differential equation.

In this paper, we mainly study subtransversality and two types of strong CHIP (given via Fréchet and limiting normal cones) for a collection of finitely many closed sets. We first prove characterizations of Asplund spaces in terms of subtransversality and intersection formulae of Fréchet normal cones. Several necessary conditions for subtransversality of closed sets are obtained via Fréchet/limiting normal cones in Asplund spaces. Then, we consider subtransversality for some special closed sets in convex-composite optimization. In this frame we prove an equivalence result on subtransversality, strong Fréchet CHIP and property (G) so as to extend a duality characterization of subtransversality of finitely many closed convex sets via strong CHIP and property (G) to the possibly non-convex case. As applications, we use these results on subtransversality and strong CHIP to study error bounds of inequality systems and give several dual criteria for error bounds via Fréchet normal cones and subdifferentials.

Stochastic embeddings of finite metric spaces into graph-theoretic trees have proven to be a vital tool for constructing approximation algorithms in theoretical computer science. In the present work, we build out some of the basic theory of stochastic embeddings in the infinite setting with an aim toward applications to Lipschitz free space theory. We prove that proper metric spaces stochastically embedding into $\\mathbb {R}$ -trees have Lipschitz free spaces isomorphic to $L^1$ -spaces. We then undergo a systematic study of stochastic embeddability of Gromov hyperbolic metric spaces into $\\mathbb {R}$ -trees by way of stochastic embeddability of their boundaries into ultrametric spaces. The following are obtained as our main results: (1) every snowflake of a compact, finite Nagata-dimensional metric space stochastically embeds into an ultrametric space and has Lipschitz free space isomorphic to $\\ell ^1$ , (2) the Lipschitz free space over hyperbolic n-space is isomorphic to the Lipschitz free space over Euclidean n-space and (3) every infinite, finitely generated hyperbolic group stochastically embeds into an $\\mathbb {R}$ -tree, has Lipschitz free space isomorphic to $\\ell ^1$ , and admits a proper, uniformly Lipschitz affine action on $\\ell ^1$ .

Let X and Y be two normed spaces. Let U be a non-principal ultrafilter on ℕ. Let g: X → Y be a standard ε -phase isometry for some ε ≥ 0, i.e., g (0) = 0, and for all u, v ϵ X , | | | | g ( u ) + g ( v ) | | ± | | g ( u ) − g ( v ) | | | − | | | u + v | | ± | | u − v | | | | ≤ ε . The mapping g is said to be a phase isometry provided that ε = 0. In this paper, we show the following universal inequality of g : for each u ∗ ∈ w ∗ − exp | | u ∗ | | B X ∗ , there exist a phase function σ u ∗ : X → { − 1 , 1 } and φ ϵ Y * with | | φ | | = | | u ∗ | | ≡ α satisfying that | 〈 u ∗ , u 〉 − σ u ∗ ( u ) 〈 φ , g ( u ) 〉 | ≤ 5 2 ε α , for all u ∈ X . In particular, let X be a smooth Banach space. Then we show the following: (1) the universal inequality holds for all u * ∈ X *; (2) the constant 5 2 can be reduced to 3 2 provided that Y * is strictly convex; (3) the existence of such a g implies the existence of a phase isometry Θ: X → Y such that Θ ( u ) = lim n , U g ( n u ) n provided that Y ** has the w *-Kadec-Klee property (for example, Y is both reflexive and locally uniformly convex).