# Weak moduli of convexity.

Extracta Mathematicae (1991)

• Volume: 6, Issue: 1, page 47-49
• ISSN: 0213-8743

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## Abstract

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Let E be a real normed linear space with unit ball B and unit sphere S. The classical modulus of convexity of J. A. Clarkson [2] δE(ε) = inf {1 - 1/2||x + y||: x,y ∈ B, ||x - y|| ≥ ε} (0 ≤ ε ≤ 2)is well known and it is at the origin of a great number of moduli defined by several authors. Among them, D. F. Cudia [3] defined the directional, weak and directional weak modulus of convexity of E, respectively, asδE(ε,g) = inf {1 - 1/2||x + y||: x,y ∈ B, g(x-y) ≥ ε}δE(ε,f) = inf {1 - 1/2 f(x,y): x,y ∈ B, ||x - y|| ≥ ε}δE(ε,f,g) = inf {1 - 1/2 f(x,y): x,y ∈ B, g(x-y) ≥ ε}where 0 ≤ ε ≤ 2 and f,g ∈ S' (unit sphere of the topological dual space E').D. F. Cudia [3] has shown the close connection existing between these moduli and various differentiability conditions of the norm in E'.In this note we study these moduli from a different point of view, then we analyze some of its properties and we see that it is possible to characterize inner product spaces by means of them.

## How to cite

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Alonso, Javier, and Ullán, Antonio. "Weak moduli of convexity.." Extracta Mathematicae 6.1 (1991): 47-49. <http://eudml.org/doc/39919>.

@article{Alonso1991,
abstract = {Let E be a real normed linear space with unit ball B and unit sphere S. The classical modulus of convexity of J. A. Clarkson [2] δE(ε) = inf \{1 - 1/2||x + y||: x,y ∈ B, ||x - y|| ≥ ε\} (0 ≤ ε ≤ 2)is well known and it is at the origin of a great number of moduli defined by several authors. Among them, D. F. Cudia [3] defined the directional, weak and directional weak modulus of convexity of E, respectively, asδE(ε,g) = inf \{1 - 1/2||x + y||: x,y ∈ B, g(x-y) ≥ ε\}δE(ε,f) = inf \{1 - 1/2 f(x,y): x,y ∈ B, ||x - y|| ≥ ε\}δE(ε,f,g) = inf \{1 - 1/2 f(x,y): x,y ∈ B, g(x-y) ≥ ε\}where 0 ≤ ε ≤ 2 and f,g ∈ S' (unit sphere of the topological dual space E').D. F. Cudia [3] has shown the close connection existing between these moduli and various differentiability conditions of the norm in E'.In this note we study these moduli from a different point of view, then we analyze some of its properties and we see that it is possible to characterize inner product spaces by means of them.},
author = {Alonso, Javier, Ullán, Antonio},
journal = {Extracta Mathematicae},
language = {eng},
number = {1},
pages = {47-49},
title = {Weak moduli of convexity.},
url = {http://eudml.org/doc/39919},
volume = {6},
year = {1991},
}

TY - JOUR
AU - Alonso, Javier
AU - Ullán, Antonio
TI - Weak moduli of convexity.
JO - Extracta Mathematicae
PY - 1991
VL - 6
IS - 1
SP - 47
EP - 49
AB - Let E be a real normed linear space with unit ball B and unit sphere S. The classical modulus of convexity of J. A. Clarkson [2] δE(ε) = inf {1 - 1/2||x + y||: x,y ∈ B, ||x - y|| ≥ ε} (0 ≤ ε ≤ 2)is well known and it is at the origin of a great number of moduli defined by several authors. Among them, D. F. Cudia [3] defined the directional, weak and directional weak modulus of convexity of E, respectively, asδE(ε,g) = inf {1 - 1/2||x + y||: x,y ∈ B, g(x-y) ≥ ε}δE(ε,f) = inf {1 - 1/2 f(x,y): x,y ∈ B, ||x - y|| ≥ ε}δE(ε,f,g) = inf {1 - 1/2 f(x,y): x,y ∈ B, g(x-y) ≥ ε}where 0 ≤ ε ≤ 2 and f,g ∈ S' (unit sphere of the topological dual space E').D. F. Cudia [3] has shown the close connection existing between these moduli and various differentiability conditions of the norm in E'.In this note we study these moduli from a different point of view, then we analyze some of its properties and we see that it is possible to characterize inner product spaces by means of them.
LA - eng
UR - http://eudml.org/doc/39919
ER -

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