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In this article, with simple and short proofs without applying fixed point theorems, some hyperstability results corresponding to the functional equations of Cauchy and Jensen are presented in 2-normed spaces. We also obtain some results on hyperstability for the general linear functional equation f (a x + b y) = A f (x) + B f (y) + C . [ABSTRACT FROM AUTHOR]

This article is devoted to introduce a new geometric constant called Dehghan–Rooin constant, which quantifies the difference between angular and skew angular distances in Banach spaces. We quantify the characterization of uniform non-squareness in terms of Dehghan–Rooin constant. The relationships between Dehghan–Rooin constant and uniform convexity, Dehghan-Rooin constant and uniform smoothness are also studied. Moreover, some new sufficient conditions for uniform normal structure are also established in terms of Dehghan–Rooin constant. [ABSTRACT FROM AUTHOR]

Let A be a unital C ∗ -algebra with unit 1 A and let a ∈ A be a positive and invertible element. Suppose that S (A) is the set of all states on A and let S a (A) = f f (a) : f ∈ S (A) , f (a) ≠ 0. The norm ‖ x ‖ a for every x ∈ A is defined by ‖ x ‖ a = sup φ ∈ S a (A) φ (x ∗ a x). In this paper, we aim to investigate the notion of Birkhoff–James orthogonality with respect to the norm ‖ · ‖ a in A , namely a-Birkhoff–James orthogonality. The characterization of a-Birkhoff–James orthogonality in A by means of the elements of generalized state space S a (A) is provided. As an application, a characterization for the best approximation to elements of A in a subspace B with respect to ‖ · ‖ a is obtained. Moreover, a formula for the distance of an element of A to the subspace B = C 1 A is given. We also study the strong version of a-Birkhoff–James orthogonality in A. The classes of C ∗ -algebras in which these two types orthogonality relationships coincide are described. In particular, we prove that the condition of the equivalence between the strong a-Birkhoff–James orthogonality and A -valued inner product orthogonality in A implies that the center of A is trivial. Finally, we show that if the (strong) a-Birkhoff–James orthogonality is right-additive (left-additive) in A , then the center of A is trivial. [ABSTRACT FROM AUTHOR]

In this paper, we shall consider a new constant D W B (X) which is the Dunkl–Williams constant related to Birkhoff orthogonality, and a constant D W B p (X) that is a generalization of D W B (X) . Interestingly, the upper bounds of D W B (X) and the Dunkl–Williams constant are different. The connections between these two constants and other well-known constants are exhibited. Some characterizations of Hilbert space and uniformly non-square Banach space in terms of these two constants are established. Furthermore, we also give a characterization of the Radon plane with an affine regular hexagonal unit sphere and calculate the value of D W B p (l ∞ - l 1) . [ABSTRACT FROM AUTHOR]

The aim of the present article is to extend the Stone-Weierstrass theorem to functions ranging in a lattice normed space and admitting order rather than topological approximation. We proceed with the machinery of Boolean-valued transfer from lattice normed space to normed space. [ABSTRACT FROM AUTHOR]

Angular equivalence of norms, introduced by Kikianty and Sinnamon (2017), is a notion of norm equivalence that is more attuned to the geometry of the norms. For certain geometrical properties and two angularly equivalent norms, it is the case that if one of the norms has a property, then so does the other. In this paper, we show further results in this direction, namely angularly equivalent norms share the property of non-squareness; and that the exposed points of the unit balls are in the same direction, under the condition that these points are assumed to be smooth with respect to both norms. A discussion on (the angular equivalence of) the dual norms of angularly equivalent norms is also given, giving a partial answer to an open problem as stated in the paper by Kikianty and Sinnamon (2017). [ABSTRACT FROM AUTHOR]

The aim of the paper is to discuss and clarify some concepts of the geometric theory of normed spaces. We mainly intend to present recent results concerning the concept of smoothness of normed spaces in connection with the concepts of the strict and uniform convexity of those spaces. [ABSTRACT FROM AUTHOR]

We study two constants g 1 (X) and J 1 (X) introduced by Fu et al. (Symmetry 13(6):951, 2021), present new characterizations of them, clarify detailed relations of these constants to James and Schäffer constants as well as the relation between J 1 (X) and A 2 (X) defined by Baronti et al. (J Math Anal Appl 252(1):124–146, 2000). We also pose several problems. [ABSTRACT FROM AUTHOR]

The geometry of the compact convex set of all n×nn\times n doubly stochastic matrices, a structure frequently referred to as the Birkhoff polytope, has been an active subject of research as of late. Geometric characteristics such as the Chebyshev center and the Chebyshev radius with respect to the operator norms from ℓnp{\ell }_{n}^{p} to ℓnp{\ell }_{n}^{p} and the Schatten pp-norms, both for the range 1≤p≤∞1\le p\le \infty , have only recently been studied in depth. In this article, we continue in this vein by determining the diameter of the Birkhoff polytope with respect to the metrics induced by the aforementioned matrix norms.

In this article, we introduce and study some strong convergence theorems for a mixed-type SP-iteration for three asymptotically nonexpansive self-mappings and three asymptotically nonexpansive nonself-mappings in uniformly convex hyperbolic spaces. In addition to that, we provide an illustrative example. The findings here expand and improve upon some of the relevant conclusions found in the published literature.

Suppose X and Y are Banach spaces, K is a compact Hausdorff space, Σ is the σ-algebra of Borel subsets of K , C (K, X) is the Banach space of all continuous X-valued functions (with the supremum norm), and T : C (K, X) → Y is a strongly bounded operator with representing measure m : Σ → L(X, Y). We show that if T is a strongly bounded operator and : B(K, X) → Y is its extension, then T* is pseudo weakly compact (resp. limited completely continuous, limited p-convergent, 1 ≤ p < ∞) if and only if has the same property. [ABSTRACT FROM AUTHOR]

In this work, we give some criteria of the weakly compact sets and a representation theorem of Riesz's type in Musielak sequence spaces using the ideas and techniques of sequence spaces and Musielak function. Finally, as an immediate consequence of the criteria considered in this paper, the criteria of the weakly compact sets of Orlicz sequence spaces are deduced. [ABSTRACT FROM AUTHOR]

In this article, we prove the existence of the best proximity point for the class of nonself generalized pseudo-contractive and Lipschitzian mappings. Also, we approximate the best proximity point through the proposed Ishikawa's iteration process for the case of nonself-mappings. Finally, we provide an example to illustrate our main result. [ABSTRACT FROM AUTHOR]

Gähler (Untersuchungen über verallgemeinerte m-metrische Räume. I". German. Math Nachr 40, 165–189, 1969) investigated m-metric spaces and in particular m-normed spaces over the field of reals. Here the existence of m-norms will be investigated and this in the more general setting of vector spaces over arbitrary non-trivial valued fields. [ABSTRACT FROM AUTHOR]

For the well developed notion of approximate Birkhoff-James orthogonality, in a real or complex normed linear space, we formulate a new characterization. It can be derived from other, already known, characterizations as well as obtained in a more elementary and direct way, on the basis of some simple inequalities for real convex functions. [ABSTRACT FROM AUTHOR]

Let X , Y be real, infinite-dimensional Banach spaces. Let ℒ ⁢ (X , Y) be the space of bounded operators. An important aspect of understanding differentiability of the operator norm at A ∈ ℒ ⁢ (X , Y) is to estimate the limit (which always exists) lim t → 0 + ⁡ ∥ A + t ⁢ B ∥ - ∥ A ∥ t for ⁢ B ∈ ℒ ⁢ (X , Y) , using the values of B on the state space S A = { τ ∈ ℒ ⁢ (X , Y) ∗ : τ ⁢ (A) = ∥ A ∥ , ∥ τ ∥ = 1 }. In this paper, we give several examples of Banach spaces, including the ℓ p spaces (for 1 < p < ∞ ) where a more tangible estimate is possible, under additional hypotheses on A. We also use the notion of norm-weak upper-semi-continuity (usc, for short) of the preduality map to achieve this. Our results also show that the operator subdifferential limit is related to the corresponding subdifferential limit of the vectors in the range space, when A ∗ ∗ attains its norm. [ABSTRACT FROM AUTHOR]

N. Gastinel and J.L. Joly defined the rectangular constant μ in Banach spaces using the notion of orthogonality according to Birkhoff and its generalization μ p , with p ≥ 1 . Recently, M. Baronti, E. Casini, and P.L. Papini defined a new constant, the isosceles constant H, in Banach spaces in a very similar way to the rectangular constant, but in this case using the isosceles orthogonality defined by James. In this paper, first of all, we generalize such constant, by defining a new constant H p that generalizes the isosceles constant H as well μ p generalizes μ . After that, we explain its properties, and we give a characterization of Hilbert spaces in terms of it. Moreover a partial characterization of uniformly non-square spaces is given. We conclude by a conjecture about the characterization of uniformly non-square spaces. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We characterise those Banach spaces X which satisfy that L(Y, X) is octahedral for every non-zero Banach space Y. They are those satisfying that, for every finite dimensional subspace Z, ℓ ∞ can be finitely-representable in a part of X kind of ℓ 1 -orthogonal to Z. We also prove that L(Y, X) is octahedral for every Y if, and only if, L (ℓ p n , X) is octahedral for every n ∈ N and 1 < p < ∞ . Finally, we find examples of Banach spaces satisfying the above conditions like Lip 0 (M) spaces with octahedral norms or L 1 -preduals with the Daugavet property. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Suppose X and Y are Banach spaces, K is a compact Hausdorff space, Σ is the σ -algebra of Borel subsets of K, C (K , X) is the Banach space of all continuous X-valued functions (with the supremum norm), and T : C (K , X) → Y is a strongly bounded operator with representing measure m : Σ → L (X , Y) . We show that if T ^ : B (K , X) → Y is its extension, then T is weak Dunford--Pettis (resp.weak* Dunford--Pettis, weak p-convergent, weak*p-convergent) if and only if T ^ has the same property. We prove that if T : C (K , X) → Y is strongly bounded limited completely continuous (resp. limited p-convergent), then m (A) : X → Y is limited completely continuous (resp. limited p-convergent) for each A ∈ Σ . We also prove that the above implications become equivalences when K is a dispersed compact Hausdorff space. [ABSTRACT FROM AUTHOR]

We study rotational surfaces with constant Minkowski Gaussian curvature and rotational surfaces with constant Minkowski mean curvature in a 3-dimensional normed space with rotationally symmetric norm. We have a generalization of the catenoid, pseudo-sphere and Delaunay surfaces. [ABSTRACT FROM AUTHOR]

In a real Banach space X, we prove that a polyhedral point is weaker than QP-point and stronger than Q-point and every weakly polyhedral Q-point is a polyhedral point. Further, we show that a norm on X is Gateaux smooth at a polyhedral point u if and only if there exists a maximal face D of the closed unit ball of X such that u is in the relative interior of D. [ABSTRACT FROM AUTHOR]

We consider weighted composition operators, that is operators of the type g ↦ w · g ∘ f , acting on spaces of Lipschitz functions. Bounded weighted composition operators, as well as some compact weighted composition operators, have been characterized quite recently. In this paper, we provide a different approach involving their pre-adjoint operators, namely the weighted Lipschitz operators acting on Lipschitz free spaces. This angle allows us to retrieve and sometimes improve some results from the literature. Notably, we obtain a distinct characterization of boundedness with a precise estimate of the norm. We also characterise injectivity, surjectivity, compactness and weak compactness in full generality. [ABSTRACT FROM AUTHOR]

In this article, a definition of a b(αn,βn){b}_{\left({\alpha }_{n},{\beta }_{n})}-best approximations of b(αn,βn){b}_{\left({\alpha }_{n},{\beta }_{n})}-hypermetric spaces over Banach algebras is given. Our objective is to prove the concept of extension of fixed point theorems in nn-ary orthogonal b(αn,βn){b}_{\left({\alpha }_{n},{\beta }_{n})}-hypermetric spaces over Banach algebras.

Abstract We study the weak convergence of infinite products of convex combinations of operators in complete CAT(0) spaces. We provide a new approach to this problem by considering a constructive selection of convex combinations in CAT(0) spaces that does not depend on the order of the involved elements and retain continuity properties with respect to them.

In this work, it is assumed that the norm over bicomplex numbers is the hyperbolic ( 픻 {\mathbb{D}} -valued) norm. In this paper, we provide an overview of bicomplex Lebesgue spaces and investigate some of their geometric properties, including 픹 ⁢ ℂ {\mathbb{B}\mathbb{C}} -convexity, 픹 ⁢ ℂ {\mathbb{B}\mathbb{C}} -strict convexity, and 픹 ⁢ ℂ {\mathbb{B}\mathbb{C}} -uniform convexity. Moreover, the basic inequalities such as 픻 {\mathbb{D}} -Hölder’s inequality and 픻 {\mathbb{D}} -Minkowski inequality for bicomplex Lebesgue spaces are presented, used to show geometric properties. [ABSTRACT FROM AUTHOR]

We study in this paper a few remarkable properties of twisted sums Z(κ, X) of c0(κ) and a Banach space X. We first prove a representation theorem for such twisted sums from which we will obtain, among others, the following: (a) twisted sums of c0(κ) and c0(I) are either subspaces of ℓ∞(κ) or contain a complemented copy of c0(κ+); (b) under the hypothesis [p = c], when K is either a suitable Corson compact, a separable Rosenthal compact or a scattered compact of finite height, there is a twisted sum of c0 and C(K) that is not isomorphic to a space of continuous functions; (c) all twisted sums Z(κ, X) are isomorphically Lindenstrauss spaces when X is a Lindenstrauss space; (d) all twisted sums Z(κ, X) are isomorphically polyhedral when X is a polyhedral space with a σ-discrete boundary, which solves a problem of Castillo and Papini. [ABSTRACT FROM AUTHOR]

Given an open subset U of C n , a weight v on U and a complex Banach space F, let H v (U , F) denote the Banach space of all weighted holomorphic mappings f : U → F , under the weighted supremum norm f v : = sup v (z) f (z) : z ∈ U. We prove that the set of all mappings f ∈ H v (U , F) that attain their weighted supremum norms is norm dense in H v (U , F) , provided that the closed unit ball of the little weighted holomorphic space H v 0 (U , F) is compact-open dense in the closed unit ball of H v (U , F). We also prove a similar result for mappings f ∈ H v (U , F) such that vf has a relatively compact range. [ABSTRACT FROM AUTHOR]

The current best algorithms for the convex body chasing (CBC) problem in online algorithms use the notion of the Steiner point of a convex set. In particular, the algorithm that always moves to the Steiner point of the request set is O(d) competitive for nested CBC, and this is optimal among memoryless algorithms [Bubeck et al.: Chasing nested convex bodies nearly optimally. In: 31st Annual ACM–SIAM Symposium on Discrete Algorithms (Salt Lake City 2020), pp. 1496–1508. SIAM, Philadelphia (2020)]. A memoryless algorithm coincides with the notion of a selector in functional analysis. The Steiner point is noted for being Lipschitz with respect to the Hausdorff metric, and for achieving the minimal Lipschitz constant possible. It is natural to ask whether every selector with this Lipschitz property yields a competitive algorithm for nested CBC. We answer this question in the negative by exhibiting a selector that yields a non-competitive algorithm for nested CBC but is Lipschitz with respect to Hausdorff distance. Furthermore, we show that being Lipschitz with respect to an L p -type analog of the Hausdorff distance is sufficient to guarantee competitiveness if and only if p = 1 . [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a new regular matrix B ~ and define the sequence space ℓ p ( B ~ ). We also examine some topological properties for this space and determine α - , β - , γ - duals. As well, we characterize some matrix classes and investigate some geometric properties such as uniform convexity, strict convexity and super-reflexivity. [ABSTRACT FROM AUTHOR]

Let Y be a uniformly convex space with power type p, and let (G , +) be an abelian group, δ , ε ≥ 0 , 0 < r < 1 . We first show a stability result for approximate isometries from an arbitrary Banach space into Y. This is a generalization of Dolinar' results for (δ , r) -isometries of Hilbert spaces and L p (1 < p < ∞ ) spaces. As a result, we prove that if a standard mapping F : G → Y satisfies d (u , F (G)) ≤ δ ∥ u ∥ r for every u ∈ Y and | ∥ F (x) − F (y) ∥ − ∥ F (x − y) ∥ | ≤ ε , x , y ∈ G , then there is an additive operator A : G → Y such that ∥ F (x) − A x ∥ = o (∥ F (x) ∥) as ∥ F (x) ∥ → ∞. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the reverse of the generalized triangle inequality of the second type in quasi-Banach spaces. More exactly, by using the concept of equivalent p-norms, we provide some necessary and sufficient conditions for n-tuples to satisfy the mentioned inequality. As applications, we improve some already known results and present some characterizations of p-Banach spaces among quasi-Banach spaces. In particular, we prove that a quasi-Banach space such as X is a p-Banach space if and only if for all (μ 1 , ⋯ , μ n) ∈ R n satisfying μ j > 0 for some j and μ i < 0 for all i ≠ j , the generalized triangle inequality of the second type ∑ i = 1 n ‖ x i ‖ p μ i ≤ ‖ ∑ i = 1 n x i ‖ p (x i ∈ X) holds only with the assumption μ j ≥ max i ∈ { 1 , ⋯ , n } \ { j } { 1 , | μ i | } . [ABSTRACT FROM AUTHOR]

We show that the algebra of cylinder functions in the Wasserstein Sobolev space H 1 , q (P p (X , d) , W p , d , m) generated by a finite and positive Borel measure m on the (p , d) -Wasserstein space (P p (X , d) , W p , d) on a complete and separable metric space (X , d) is dense in energy. As an application, we prove that, in case the underlying metric space is a separable Banach space B , then the Wasserstein Sobolev space is reflexive (resp. uniformly convex) if B is reflexive (resp. if the dual of B is uniformly convex). Finally, we also provide sufficient conditions for the validity of Clarkson's type inequalities in the Wasserstein Sobolev space. [ABSTRACT FROM AUTHOR]