The Dunkl–Williams constant, convexity, smoothness and normal structure

Jiménez-Melado, Antonio; Llorens-Fuster, Enrique; Mazcuñán-Navarro, Eva M.

doi:10.1016/j.jmaa.2007.11.045

Cited by 21 publications

(14 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We show that IB(X) > 1/2 if and only if the space X is uniformly non-square. To do this, we need to recall the Dunkl-Williams constant defined in [8]:…”

Section: Thus We Havementioning

confidence: 99%

The constants to measure the differences between Birkhoff and isosceles orthogonalities

Mizuguchi

2016

Filomat

View full text Add to dashboard Cite

The notion of orthogonality for vectors in inner product spaces is simple, interesting and fruitful.When moving to normed spaces, we have many possibilities to extend this notion. We consider Birkhoff orthogonality and isosceles orthogonality, which are the most used notions of orthogonality. In 2006, Ji and Wu introduced a geometric constant D(X) to give a quantitative characterization of the difference between these two orthogonality types. However, this constant was considered only in the unit sphere S X of the normed space X. In this paper, we introduce a new geometric constant IB(X) to measure the difference between Birkhoff and isosceles orthogonalities in the entire normed space X. To consider the difference between these orthogonalities, we also treat constant BI(X). ∀λ ∈ R, x + λy = x − λy .Birkhoff [3] introduced Birkhoff orthogonality: x is said to be Birkhoff orthogonal to y (denoted byJames [5] introduced isosceles orthogonality: x is said to be isosceles orthogonal to y (denoted by x ⊥ I y) ifThese generalized orthogonality types have been studied in a lot of papers ([1, 6] and so on).

show abstract

“…We show that IB(X) > 1/2 if and only if the space X is uniformly non-square. To do this, we need to recall the Dunkl-Williams constant defined in [8]:…”

Section: Thus We Havementioning

confidence: 99%

The constants to measure the differences between Birkhoff and isosceles orthogonalities

Mizuguchi

2016

Filomat

View full text Add to dashboard Cite

show abstract

“…Then the frame of B X is defined by frm(B X ) = {E( f ) : f is a support functional for B X }. This was first introduced in [18] to construct a new calculation method for the Dunkl-Williams constant (compare [14]). Its geometric and topological properties were studied in [21].…”

Section: Preliminariesmentioning

confidence: 99%

A Further Property of Spherical Isometries

Tanaka¹

2014

Bull. Aust. Math. Soc.

View full text Add to dashboard Cite

In this paper, it is proved that every isometry between the unit spheres of two real Banach spaces preserves the frames of the unit balls. As a consequence, if X and Y are n-dimensional Banach spaces and T 0 is an isometry from the unit sphere of X onto that of Y then it maps the set of all (n − 1)-extreme points of the unit ball of X onto that of Y.2010 Mathematics subject classification: primary 46B04; secondary 46B20.

show abstract

“…In [4], three geometric constants have been computed for Morrey spaces. The first two constants, namely Von Neumann-Jordan constant and James constant, are closely related to the notion of uniformly non-squareness of (the unit ball in) a Banach space [5,6,8]. For a general Banach space ( , ‖ • ‖ ), the constants are defined by Note also that, for 1 ∞, we have [2,3]:…”

Section: Introductionmentioning

confidence: 99%

On geometric properties of Morrey spaces

2021

View full text Add to dashboard Cite

The study of Morrey spaces is motivated by many reasons. Initially, these spaces were introduced in order to understand the regularity of solutions to elliptic partial differential equations [1]. In line with this, many authors study the boundedness of various integral operators on Morrey spaces. In this article, we are interested in their geometric properties, from functional analysis point of view. We show constructively that Morrey spaces are not uniformly non-ℓ 1 for any 2. This result is sharper than earlier results, which showed that Morrey spaces are not uniformly non-square and also not uniformly nonoctahedral. We also discuss the -th James constant ( ) J ( ) and the -th Von Neumann-Jordan constant ( ) NJ ( ) for a Banach space , and obtain that both constants for any Morrey space ℳ (R ) with 1 < < ∞ are equal to .

show abstract

The Dunkl–Williams constant, convexity, smoothness and normal structure

Cited by 21 publications

References 11 publications

The constants to measure the differences between Birkhoff and isosceles orthogonalities

The constants to measure the differences between Birkhoff and isosceles orthogonalities

A Further Property of Spherical Isometries

On geometric properties of Morrey spaces

Contact Info

Product

Resources

About