More on the (Uniform) Mazur Intersection Property

In this paper, we introduce two moduli of w*-semidenting points and characterise the Mazur Intersection Property (MIP) and the Uniform MIP (UMIP) in terms of these moduli. We show that a property slightly stronger than UMIP already implies uniform convexity of the dual. This may lead to a possible approach towards answering the long standing open question whether the UMIP implies the existence of an equivalent uniformly convex renorming. We also obtain the condition for stability of the UMIP under $\ell_p$-sums.