Mazur Intersection Properties and Differentiability of Convex Functions in Banach Spaces

Abstract

AIt is proved that the dual of a Banach space with the Mazur intersection property is almost weak* Asplund. Analogously, the predual of a dual space with the weak* Mazur intersection property is almost Asplund. Through the use of these arguments, it is found that, in particular, almost all (in the Baire sense) equivalent norms on % "(Γ) and % _ (Γ) are Fre! chet differentiable on a dense G δ subset. Necessary conditions for Mazur intersection properties in terms of convex sets satisfying a Krein-Milman type condition are also discussed. It is also shown that, if a Banach space has the Mazur intersection property, then every subspace of countable codimension can be equivalently renormed to satisfy this property.

A note on porosity and the Mazur intersection property

“…More precisely, let // be the collection of all intersections of balls, considered as a subset of the hyperspace ft of all closed, convex and bounded sets of a Banach space, furnished with the Hausdorff metric. We prove that // is uniformly very porous if and only if the space fails the Mazur intersection property, thus improving a result obtained in [6]. Actually, in this case, // turns out to be porous in a much stronger sense, somehow close to the notion of cone-meager introduced by Preiss and Zajicek [13], [14].…”

Antiproximinal Norms in Banach Spaces

“…This assertion can be proved by observing that the set of dual norms which are Fréchet differentiable at f is residual. The proof of this fact is analogous to the one given in [16,26]. Nevertheless, it is unknown whether the set of norms admitting a companion is first Baire category in N(X).…”

“…Then 4 n X g n … NA || · || thus implying that NA || · || is residual. Relatedly, in locally uniformly rotund Banach spaces, every locally uniformly rotund body C has a dense set of nearest points in "C (see [17] for the definition of locally uniformly rotund body). This is a consequence of the fact that the set of common support functionals to C and the unit ball is G d and dense and the norm topology in "C is given by slices.…”

External analysis of boundary points of convex sets: illumination and visibility

“…To see that x is continuous, consider a reference point z ∈ int(K) and a sequence {z n } n∈N → z. For each n ∈ N, there is a unique t n ∈ ]0, 1[ such that (7) x (z n ) = x + t n (z n − x).…”

“…It has been traditionally considered only as a matter for norms and, consequently, closed balls. However, the definition can be easily generalized for Minkowski's gauges, as done in [7], and arbitrary convex sets, as done in the above definition. Lemma 3.9.…”