Weak moduli of convexity.

Alonso, Javier ; Ullán, Antonio

Extracta Mathematicae, Tome 6 (1991), p. 47-49 / Harvested from Biblioteca Digital de Matemáticas

Résumé

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, as

δE(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.

Détail

Publié le : 1991-01-01
DMLE-ID : 2591