Intersections of Closed Balls and Geometry of Banach Spaces

In section 1 we present definitions and basic results concerning the Mazur intersection property (MIP) and some of its related properties as the MIP* . Section 2 is devoted to renorming Banach spaces with MIP and MIP*. Section 3 deals with the connections between MIP, MIP* and differentiability of convex functions. In particular, we will focuss on Asplund and almost Asplund spaces. In Section 4 we discuss the interplay between porosity and MIP. Finally, in section 5 we are concerned with the stability of the (closure of the) sum of convex sets which are intersections of balls and with Mazur spaces. 1. The Mazur intersection property and its relatives It was Mazur [39] who first drew attention to the euclidean space property: every bounded closed convex set can be represented as an intersection of closed balls. He began the investigation to determine those normed linear spaces which posses this property, named after him the Mazur intersection property or MIP. He proved Theorem 1.1, whose proof is so nice and clear that it deserves to be the starting point for this survey. The following easy (and useful) fact will be used extensively throughout the rest of the paper: a closed, convex and bounded set C is an intersection of balls if and only if for every x / ∈ C, there is a closed ball containing the set but missing the point. Hence, the MIP can be regarded as a separation property by balls which is stronger than the classical separation property by hyperplanes. We denote by B and S the unit ball and unit sphere of a Banach space. Analogously, B∗ and S∗ will stand for the corresponding unit ball and unit sphere in the dual space. Theorem 1.1. If a norm ‖·‖ in a Banach space X is Fréchet differentiable, then (X, ‖·‖) satisfies the Mazur intersection property. Proof. Consider a closed convex and bounded set C and assume that 0 / ∈ C. We will find x ∈ X and r > 0 such that C ⊂ x + rB but 0 / ∈ (x + rB). Since 0 / ∈ C, there is 1991 Mathematics Subject Classification. 46B20.