ON A GENERALIZED MODULUS OF CONVEXITY AND UNIFORM NORMAL STRUCTURE

In this article, the authors study a generalized modulus of convexity, δ（α） （ε）. Certain related geometrical properties of this modulus are analyzed. Their main result is that Banach space X has uniform normal structure if there exists ε, 0 ≤e≤ 1, such that δ^（α）（1＋ε ） 〉 （1- α）ε.

NORMAL STRUCTURE AND SOME GEOMETRIC PARAMETERS RELATED TO THE MODULUS OF U-CONVEXITY IN BANACH SPACES

We present a short and simple proof of the recent result of Yang and Wang [12]. Stimulated by their idea, two geometric parameters Uax（ε） and βx （ε）, both related to Gao＇s modulus of U-convexity of a Banach space X, are introduced. Their properties and the relationships with normal structure are studied. Some existing results involving normal structure and fixed points for non-expansive mappings in Banach spaces are improved.

ON A CLASS OF KTHE SEQUENCE SPACES WITH NORMAL STRUCTURE

In this note, we investigate the generalized modulus of convexity δ（λ） and the generalized modulus smoothness p（λ）. We obtain some estimates of these moduli for X=lp. We obtain inequalities between WCS coefficient of a KSthe sequence space X and δX（λ）. We show that, for a wide class of class the sequence spaces X, if for some ε∈（0, 9/10] holds δx（e） 〉 1/3（1- √3/2）ε, then X has normal structure.

A New Class of Banach Spaces with Uniform Normal Structure

Let X be a Banach space, S(X) be the unit sphere of X, φ be a function: S(X)→ S(X *) such that φ(x)∈ x, and v φ(ε) =inf 1-12x+y: x,y∈S(X), and 〈φ(x), x-y 〉≥ε, 0≤ε≤2, where x is the set of norm 1 supporting functionals of S(X) at x. A geometric concept, modulus of V convexity V(ε)= sup {V φ(ε), for all φ: S(X)→S(X *)}, is introduced; the properties of V(ε) and the relationship between V(ε) and other geometric concepts are discussed. The main result is that V12>0 implies normal structure.

On a New Geometric Constant Related to the Modulus of Smoothness of a Banach Space

We shall introduce a new geometric constant A（X） of a Banach space X, which is closely related to the modulus of smoothness px （T）, and investigate it in relation with the constant As （X） by Baronti et al., the von Neumann-Jordan constant CNj（X） and the James constant J（X）. A sequence of recent results on these constants as well as some other geometric constants will be strengthened and improved.

A Note of Paper ＂Banach Spaces Failing the Almost Isometric Universal Extension Property＂

The definition of property A with constant α was introduced by D. M. Speegle, who proved that every infinite dimensional separable uniformly smooth Banach space has property A with constant α∈ [0, 1）. In this paper, we give a sufficient condition for a Banach space to have property A with constant α∈[0, 1）, and some remarks on Speegle＇s paper .

The James Constant for the l_3-l_1 Space

In this note, the exact value of the James constant for the l3 - l1 space is obtained, J（l3 - l0 = 1.5573.... This result improves the known inequality, J（13 - 11） ≤4/3√10,which was given by Dhompongsa, Piraisangjun and Saejung.