More on the (uniform) Mazur intersection property

We introduce two moduli of $w^*$-semidenting points and characterise the Mazur Intersection Property (MIP) and the Uniform Mazur Intersection Property (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 the long standing open question whether UMIP implies the existence of an equivalent uniformly convex renorming.

We also obtain a condition for stability of UMIP under $\ell_p$-sums.