scholarly journals Geometric Constants in Banach Spaces Related to the Inscribed Quadrilateral of Unit Balls

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1294
Author(s):  
Asif Ahmad ◽  
Qi Liu ◽  
Yongjin Li
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Normal Structure ◽  
Inner Product ◽  
Sufficient Condition ◽  
Basic Properties ◽  
Geometric Constant ◽  
Geometric Constants

We introduce a new geometric constant Jin(X) based on a generalization of the parallelogram law, which is symmetric and related to the length of the inscribed quadrilateral side of the unit ball. We first investigate some basic properties of this new coefficient. Next, it is shown that, for a Banach space, Jin(X) becomes 16 if and only if the norm is induced by an inner product. Moreover, its properties and some relations between other well-known geometric constants are studied. Finally, a sufficient condition which implies normal structure is presented.

scholarly journals Some aspects of generalized Zbăganu and James constant in Banach spaces

2021 ◽  
Vol 54 (1) ◽  
pp. 299-310
Author(s):  
Qi Liu ◽  
Muhammad Sarfraz ◽  
Yongjin Li
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Banach Spaces ◽  
Normal Structure ◽  
Inner Product ◽  
Sufficient Condition ◽  
Point Property ◽  
James Constant ◽  
Geometric Constant

Abstract We shall introduce a new geometric constant C Z ( λ , μ , X ) {C}_{Z}\left(\lambda ,\mu ,X) based on a generalization of the parallelogram law, which was proposed by Moslehian and Rassias. First, it is shown that, for a Banach space, C Z ( λ , μ , X ) {C}_{Z}\left(\lambda ,\mu ,X) is equal to 1 if and only if the norm is induced by an inner product. Next, a characterization of uniformly non-square is given, that is, X X has the fixed point property. Also, a sufficient condition which implies weak normal structure is presented. Moreover, a generalized James constant J ( λ , X ) J\left(\lambda ,X) is also introduced. Finally, some basic properties of this new coefficient are presented.

scholarly journals On a New Geometric Constant Related to the Euler-Lagrange Type Identity in Banach Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 116
Author(s):  
Qi Liu ◽  
Yongjin Li
Keyword(s):  
Banach Spaces ◽  
Product Space ◽  
Sufficient Conditions ◽  
Normal Structure ◽  
The Other ◽  
Inner Product ◽  
Equivalent Characterization ◽  
Geometric Constant ◽  
Geometric Constants

In this paper, we will introduce a new geometric constant LYJ(λ,μ,X) based on an equivalent characterization of inner product space, which was proposed by Moslehian and Rassias. We first discuss some equivalent forms of the proposed constant. Next, a characterization of uniformly non-square is given. Moreover, some sufficient conditions which imply weak normal structure are presented. Finally, we obtain some relationship between the other well-known geometric constants and LYJ(λ,μ,X). Also, this new coefficient is computed for X being concrete space.

scholarly journals New Geometric Constants in Banach Spaces Related to the Inscribed Equilateral Triangles of Unit Balls

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 951
Author(s):  
Yuankang Fu ◽  
Qi Liu ◽  
Yongjin Li
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Normal Structure ◽  
Upper And Lower Bounds ◽  
Uniformly Convex ◽  
Equilateral Triangles ◽  
Geometric Constant ◽  
Geometric Constants

Geometric constant is one of the important tools to study geometric properties of Banach spaces. In this paper, we will introduce two new geometric constants JL(X) and YJ(X) in Banach spaces, which are symmetric and related to the side lengths of inscribed equilateral triangles of unit balls. The upper and lower bounds of JL(X) and YJ(X) as well as the values of JL(X) and YJ(X) for Hilbert spaces and some common Banach spaces will be calculated. In addition, some inequalities for JL(X), YJ(X) and some significant geometric constants will be presented. Furthermore, the sufficient conditions for uniformly non-square and normal structure, and the necessary conditions for uniformly non-square and uniformly convex will be established.

scholarly journals Modulus of Convexity, the CoeffcientR(1,X), and Normal Structure in Banach Spaces

2008 ◽  
Vol 2008 ◽  
pp. 1-5 ◽  
Author(s):  
Hongwei Jiao ◽  
Yunrui Guo ◽  
Fenghui Wang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Normal Structure ◽  
Geometric Parameters ◽  
Sufficient Condition ◽  
Modulus Of Convexity

LetδX(ϵ)andR(1,X)be the modulus of convexity and the Domínguez-Benavides coefficient, respectively. According to these two geometric parameters, we obtain a sufficient condition for normal structure, that is, a Banach spaceXhas normal structure if2δX(1+ϵ)>max{(R(1,x)-1)ϵ,1-(1-ϵ/R(1,X)-1)}for someϵ∈[0,1]which generalizes the known result by Gao and Prus.

scholarly journals Some Properties Concerning the JL(X) and YJ(X) Which Related to Some Special Inscribed Triangles of Unit Ball

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1285
Author(s):  
Asif Ahmad ◽  
Yuankang Fu ◽  
Yongjin Li
Keyword(s):  
Unit Ball ◽  
Banach Spaces ◽  
Uniformly Convex ◽  
Geometric Properties ◽  
James Constant ◽  
Von Neumann ◽  
Geometric Constants

In this paper, we will make some further discussions on the JL(X) and YJ(X) which are symmetric and related to the side lengths of some special inscribed triangles of the unit ball, and also introduce two new geometric constants L1(X,▵), L2(X,▵) which related to the perimeters of some special inscribed triangles of the unit ball. Firstly, we discuss the relations among JL(X), YJ(X) and some geometric properties of Banach spaces, including uniformly non-square and uniformly convex. It is worth noting that we point out that uniform non-square spaces can be characterized by the side lengths of some special inscribed triangles of unit ball. Secondly, we establish some inequalities for JL(X), YJ(X) and some significant geometric constants, including the James constant J(X) and the von Neumann-Jordan constant CNJ(X). Finally, we introduce the two new geometric constants L1(X,▵), L2(X,▵), and calculate the bounds of L1(X,▵) and L2(X,▵) as well as the values of L1(X,▵) and L2(X,▵) for two Banach spaces.

The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

2021 ◽  
pp. 1-17
Author(s):  
Dongni Tan ◽  
Xujian Huang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Phase Function ◽  
Linear Isometry ◽  
Basic Properties ◽  
Finite Dimensional ◽  
Image Position

Abstract We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

scholarly journals A note on best approximation and invertibility of operators on uniformly convex Banach spaces

1991 ◽  
Vol 14 (3) ◽  
pp. 611-614 ◽  
Author(s):  
James R. Holub
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Best Approximation ◽  
Bounded Linear Operator ◽  
Convex Banach Space ◽  
Sufficient Condition ◽  
Uniformly Convex ◽  
Bounded Linear ◽  
Uniformly Convex Banach Spaces

It is shown that ifXis a uniformly convex Banach space andSa bounded linear operator onXfor which‖I−S‖=1, thenSis invertible if and only if‖I−12S‖<1. From this it follows that ifSis invertible onXthen either (i)dist(I,[S])<1, or (ii)0is the unique best approximation toIfrom[S], a natural (partial) converse to the well-known sufficient condition for invertibility thatdist(I,[S])<1.

scholarly journals Geometric Invariants of Surjective Isometries between Unit Spheres

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2346
Author(s):  
Almudena Campos-Jiménez ◽  
Francisco Javier García-Pacheco
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Geometric Property ◽  
The Other ◽  
Geometric Invariants ◽  
Finite Dimensional ◽  
Surjective Isometry ◽  
Finite Dimensional Case

In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let X,Y be Banach spaces and let T:SX→SY be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if F is a maximal face of the unit ball containing inner points, then T(−F)=−T(F). We also show that if [x,y] is a non-trivial segment contained in the unit sphere such that T([x,y]) is convex, then T is affine on [x,y]. As a consequence, T is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property P, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property P.

Γ‎-Null and Γ‎N-Null Sets

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Finite Dimension ◽  
Baire Category ◽  
Borel Sets ◽  
Basic Properties ◽  
Gâteaux Differentiability ◽  
Gateaux Differentiability ◽  
Null Set

This chapter introduces the notions of Γ‎-null and Γ‎ₙ-null sets, which are σ‎-ideals of subsets of a Banach space X. Γ‎-null set is key for the strongest known general Fréchet differentiability results in Banach spaces, whereas Γ‎ₙ-null set presents a new, more refined concept. The reason for these notions comes from an (imprecise) observation that differentiability problems are governed by measure in finite dimension, but by Baire category when it comes to behavior at infinity. The chapter first relates Γ‎-null and Γ‎ₙ-null sets to Gâteaux differentiability before discussing their basic properties. It then describes Γ‎-null and Γ‎ₙ-null sets of low Borel classes and presents equivalent definitions of Γ‎ₙ-null sets. Finally, it considers the separable determination of Γ‎-nullness for Borel sets.

Banach Spaces that are Uniformly Rotund in Weakly Compact Sets of Directions

1977 ◽  
Vol 29 (5) ◽  
pp. 963-970 ◽  
Author(s):  
Mark A. Smith
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Nonzero Element ◽  
Bounded Subset ◽  
Compact Sets ◽  
Weakly Compact ◽  
Uniform Rotundity ◽  
Minimum Depth ◽  
Weakly Compact Sets

In a Banach space, the directional modulus of rotundity, δ (ϵ, z), measures the minimum depth at which the midpoints of all chords of the unit ball which are parallel to z and of length at least ϵ are buried beneath the surface. A Banach space is uniformly rotund in every direction (URED) if δ (ϵ, z) is positive for every positive ϵ and every nonzero element z. This concept of directionalized uniform rotundity was introduced by Garkavi [6] to characterize those Banach spaces in which every bounded subset has at most one Čebyšev center.

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