It was proved recently that a Banach space fails the Mazur
intersection property if and only if the family of all closed,
convex and bounded subsets which are intersections of balls is
uniformly very porous. This paper deals with the geometrical
implications of this result. It is shown that every equivalent
norm on the space can be associated in a natural way with a
constant of porosity, whose interplay with the geometry of the
space is then investigated. Among other things, we prove that
this constant is closely related to the set of
$\varepsilon$-differentiability points of the space and the
set of $r$-denting points of the dual. We also obtain
estimates for this constant in several classical spaces.