# Weak moduli of convexity.

Extracta Mathematicae (1991)

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

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topAlonso, 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},

keywords = {Espacios normados; Espacios con producto interno; Convexidad; Módulo de convexidad; modulus of convexity},

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

KW - Espacios normados; Espacios con producto interno; Convexidad; Módulo de convexidad; modulus of convexity

UR - http://eudml.org/doc/39919

ER -

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