On New Moduli Related to the Generalization of the Parallelogram Law

Liu, Qi; Zhou, Chuanjiang; Sarfraz, Muhammad; Li, Yongjin

doi:10.1007/s40840-021-01196-7

Cited by 5 publications

(4 citation statements)

References 23 publications

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“…Characterizations of inner product spaces with the help of strongly convex functions can be found in Nicodem and Pales [8]. Recent papers that study conditions which imply that norms are induced by inner products include Ahmad et al [9], Cichon [10], Harandi et al [11], Krnic and Minculete [12,13] and Liu et al [14].…”

Section: Introductionmentioning

confidence: 99%

Inequalities That Imply the Norm of a Linear Space Is Induced by an Inner Product

Rădulescu,

Bencze

2023

Mathematics

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The aim of this paper is to investigate when a linear normed space is an inner product space. Several conditions in a linear normed space are formulated with the help of inequalities. Some of them are from the literature and others are new. We prove that these conditions are equivalent with the fact that the norm is induced by an inner product. One of the new results is the following: in an inner product space, the sum of opposite edges of a tetrahedron are the sides of an acute angled triangle. The converse of this result holds also. More precisely, this property characterizes inner product spaces. Another new result is the following: in a tetrahedron, the sum of squares of opposite edges are the lengths of a triangle. We prove also that this property characterizes inner product spaces. In addition, we give simpler proofs to some theorems already known from the publications of other authors.

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Section: Introductionmentioning

confidence: 99%

Inequalities That Imply the Norm of a Linear Space Is Induced by an Inner Product

Rădulescu,

Bencze

2023

Mathematics

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“…After the appearance of these constants, new geometric constants defined by scholars continue to appear in our field of vision. For more papers on geometric constants, please refer to [10,11,15]. These constants become a simple and intuitive tool for digitally quantifying the properties of Banach spaces.…”

Section: Introductionmentioning

confidence: 99%

Skew geometric mean constants in Banach spaces

Ni,

Liu,

Zhou

2023

Preprint

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In this paper, a new skew geometric mean constant is introduced first, and then another new constant is introduced after restricting the isosceles orthogonal condition. These constants are used to characterize Hilbert spaces. Some basic properties of these constants in Banach spaces are derived, and the values of constants in specific spaces are calculated. On this basis, the relationship between the new geometric constants and other famous constants is studied. Finally, based on these identities, the relationship between the new geometric constants and the geometrical properties in Banach spaces is discussed, such as uniform non-square and normal structure.

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“…Further research concerning J(X) has been conducted on the issue of whether the space has a normal structure or not. For the result and application of this constant, please refer to the article [1][2][3][4][5].…”

Section: Introductionmentioning

confidence: 99%

Inscribed Triangles in the Unit Sphere and a New Class of Geometric Constants

Bing-ren¹,

Liu²,

Li³

2022

Symmetry

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In this paper, we firstly investigate the constant H(X) proposed by Gao further by discussing several properties of it that have not yet been discovered. Secondly, we focus on a new constant GL(X) closely related to H(X), along with a variety of geometric properties. In addition, we show several relations among it and the several basic geometric constants via a few inequalities. Finally, we manage to characterize the geometric properties of its generalized forms GL(X,p) and CL(X) explicitly.

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On New Moduli Related to the Generalization of the Parallelogram Law

Cited by 5 publications

References 23 publications

Inequalities That Imply the Norm of a Linear Space Is Induced by an Inner Product

Inequalities That Imply the Norm of a Linear Space Is Induced by an Inner Product

Skew geometric mean constants in Banach spaces

Inscribed Triangles in the Unit Sphere and a New Class of Geometric Constants

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