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.

QUANTUM INFORMATION GEOMETRY AND NONCOMMUTATIVE Lp-SPACES

2005 ◽  
Vol 08 (02) ◽  
pp. 215-233 ◽  
Author(s):  
ANNA JENČOVÁ
Keyword(s):  
Quantum Information ◽  
Banach Spaces ◽  
Von Neumann Algebra ◽  
Classical Case ◽  
Information Geometry ◽  
Positive Cone ◽  
Uniformly Convex ◽  
Von Neumann ◽  
Lp Spaces ◽  
Uniformly Convex Banach Spaces

Let M be a von Neumann algebra. We define the noncommutative extension of information geometry by embeddings of M into noncommutative Lp-spaces. Using the geometry of uniformly convex Banach spaces and duality of the Lp and Lq spaces for 1/p +1/q =1, we show that we can introduce the α-divergence, for α∈(-1, 1), in a similar manner as Amari in the classical case. If restricted to the positive cone, the α-divergence belongs to the class of quasi-entropies, defined by Petz.

scholarly journals Hölder’s means and triangles inscribed in a semicircle in Banach spaces

Filomat ◽  
2012 ◽  
Vol 26 (2) ◽  
pp. 371-377
Author(s):  
Huanhuan Cui ◽  
Ge Lu
Keyword(s):  
Banach Spaces ◽  
Geometric Properties ◽  
The Stability ◽  
Geometric Constants

By the H?lder?s means, we introduce two classes geometric constants for Banach spaces. We study some geometric properties related to these constants and the stability under norm perturbations of them.

scholarly journals Some Aspects of Geometric Constants in Modular Spaces

2021 ◽  
Vol 1 (2) ◽  
pp. 151-163
Author(s):  
Zhijian Yang ◽  
Qi Liu ◽  
Muhammad Sarfraz ◽  
Yongjin Li
Keyword(s):  
Banach Spaces ◽  
Strict Convexity ◽  
Uniform Convexity ◽  
Normed Spaces ◽  
James Constant ◽  
Modular Spaces ◽  
The Relationship ◽  
Geometric Constants

In this paper, we generalize the typical geometric constants of Banach spaces to modular spaces. We study the equivalence between the convexity of modular and normed spaces, and obtain the relationship between ρ-Neumann-Jordan constant and ρ-James constant. In particular, we extend the convexity and smoothness modular, and obtain the criterion theorems of the uniform convexity and strict convexity.

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 Hölder’s means and fixed points for multivalued nonexpansive mappings

Filomat ◽  
2018 ◽  
Vol 32 (19) ◽  
pp. 6531-6547
Author(s):  
Mina Dinarvand
Keyword(s):  
Fixed Points ◽  
Banach Spaces ◽  
Recent Literature ◽  
Nonexpansive Mappings ◽  
Convergent Sequence ◽  
Normal Structure ◽  
James Constant ◽  
Von Neumann ◽  
Geometric Conditions ◽  
Weakly Convergent Sequence Coefficient

In this paper, we show some geometric conditions on Banach spaces by considering H?lder?s means and many well known parameters namely the James constant, the von Neumann-Jordan constant, the weakly convergent sequence coefficient, the normal structure coefficient, the coefficient of weak orthogonality, which imply the existence of fixed points for multivalued nonexpansive mappings and normal structure of Banach spaces. Some of our main results improve and generalize several known results in the recent literature on this topic. We also show that some of our results are sharp.

scholarly journals On Geometric Constants for Discrete Morrey Spaces

2021 ◽  
Vol 2 ◽  
pp. 2
Author(s):  
Adam Adam ◽  
Hendra Gunawan
Keyword(s):  
Morrey Spaces ◽  
James Constant ◽  
Von Neumann ◽  
Geometric Constants

In this paper we prove that the n-th Von Neumann-Jordan constant and the n-th James constant for discrete Morrey spaces lpq where 1≤p<q<∞ are both equal to n. This result tells us that the discrete Morrey spaces are not uniformly non-l1, and hence they are not uniformly n-convex.

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 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 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.

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