Abstract
In this paper, we further studied properties of the modulus of n -dimensional U -convexity and the modulus of n -dimensional U -flatness when n = 1 (2-dimensional character) and n = 2 (3-dimensional character). The new properties of these moduli are investigated, and the relationships between these moduli and other geometric parameters of Banach spaces are studied. Some results on fixed point theory for nonexpansive mappings and normal structure in Banach spaces are obtained.
Highlights
Brodskiı and Mil’man [1] introduced the following geometric concepts in 1948.Definition 1
(c) Uniform normal structure if there exists 0 < c < 1 such that, for every bounded closed convex subset C of K that contains more than one point, there is a point x0 ∈ C such that sup x0 − y : y ∈ C < c · diam C
We studied further properties of the modulus of n-dimensional U-convexity and the modulus of n-dimensional U-flatness when n 1 (2-dimensional character) and n 2 (3-dimensional character). e new properties of these moduli are investigated and the relationships between these moduli and other geometric parameters of Banach spaces are studied
Summary
Brodskiı and Mil’man [1] introduced the following geometric concepts in 1948.Definition 1. (b) If X is a Banach space with UnX(1) > 0 for some n ∈ N, X is super-reflexive.
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