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

Let X be a Banach space with the closed unit ball 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 X , and the modulus of UKK* on the dual space X ∗ of X . 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 ∗ ) is weak* sequentially compact and UKK∗ ( ( 1 μ ( X ∗ ) ) − ) > 1 − 1 μ ( X ∗ ) for X ∗ , then X has weak normal structure, where μ ( X ) is the separation measure of B ( X ) .