  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.
