Constante rectangle et biais d'un espace de Banach

Let X be a real Banach space. The rectangular constant $\mu (X)$ and some generalisations of it, $\mu _p(X)$ for $p \geq 1$ , were introduced by Gastinel and Joly around half a century ago. In this paper we make precise some characterisations of inner product spaces by using $\mu _p(X)$ , correcting some statements appearing in the literature, and extend to $\mu _p(X)$ some characterisations of uniformly nonsquare spaces, known only for $\mu (X)$ . We also give a characterisation of two-dimensional spaces with hexagonal norms. Finally, we indicate some new upper estimates concerning $\mu (l_p)$ and $\mu _p(l_p)$ .

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“…These results also follow from Theorem 3.1. Some bounds for l p spaces are given in [5]: μ(l p ) ≤ (5…”

Section: Estimates In L P Spacesmentioning

confidence: 99%

“…In Section 4 we will give a characterisation of two-dimensional spaces with symmetric orthogonality by using the parameter μ p (X) and, finally, in the last section we will improve some upper bounds obtained in [5] for the parameter μ(l p ).…”

Section: Introductionmentioning

confidence: 99%

Revisiting the Rectangular Constant in Banach Spaces

Baronti

Casini

Papini

2021

Bull. Aust. Math. Soc.

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“…The notion of rectangular constant plays a very important role in the geometry of normed linear spaces and it has been studied by Joly [8], del Rio and Benitez [3], Desbiens [4] and Baronti and Casini [1]. In 1999 the concept of the rectangular constant of a normed linear space was generalized by Serb [9] to introduce rectangular modulus, µ X (λ), as a function µ X : (0, ∞) −→ R defined as follows…”

Section: Let (Xmentioning

confidence: 99%

On rectangular constant in normed linear spaces

Paul,

Ghosh,

Sain

2014

Preprint

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We study the properties of rectangular constant µ(X) in a normed linear space X. We prove that µ(X) = 3 iff the unit sphere contains a straight line segment of length 2. In fact, we prove that the rectangular modulus attains its upper bound iff the unit sphere contains a straight line segment of length 2. We prove that if the dimension of the space X is finite then µ(X) is attained. We also prove that a normed linear space is an inner product space iff we have sup{ 1+|t|2010 Mathematics Subject Classification. Primary 46B20, Secondary 47A30. Key words and phrases. Birkhoff-James Orthogonality, rectangular constant.The second and third author would like to thank CSIR and UGC respectively for the financial support.

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Optimal Sunny Selections for Metric Projections onto Unit Balls

Hsiao

Smarzewski

1995

Journal of Approximation Theory

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Constante rectangle et biais d'un espace de Banach

Cited by 3 publications

References 17 publications

Revisiting the Rectangular Constant in Banach Spaces

Revisiting the Rectangular Constant in Banach Spaces

On rectangular constant in normed linear spaces

Optimal Sunny Selections for Metric Projections onto Unit Balls

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