A constant of porosity for convex bodies

Abstract

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.