Normal structure and moduli of UKK, NUC, and UKK* in Banach spaces

Let XX be a Banach space with the closed unit ball B(X)B(X). In this paper, by directly extrapolating from the definitions of Uniformly Kadec–Klee (UKK), Nearly Uniformly Convex (NUC) and Weak* Uniformly Kadec–Klee (w*UKK) spaces, we consider the concepts of the modulus of UKK and the modulus of NUC on XX, and the modulus of UKK* on the dual space X∗X∗ of XX. Some new properties of Banach spaces related to reflexivity and normal structure with the values of these moduli are obtained. Among these new results, we prove that if B(X∗)B(X∗) is weak* sequentially compact and UKK∗((1μ(X∗))−)>1−1μ(X∗) for X∗X∗, then XX has weak normal structure, where μ(X)μ(X) is the separation measure of B(X)B(X).