Simon Fitzpatrick

Existence of nearest points in Banach spaces

This paper makes a unified development of what the authors know about the existence of nearest points to closed subsets of (real) Banach spaces. Our work is made simpler by the methodical use of subderivatives. The results of Section 3 and Section 7 in particular are, to the best of our knowledge, new. In Section 5 and Section 6 we provide refined proofs of the Lau-Konjagin nearest point characterizations of reflexive Kadec spaces (Theorem 5.11, Theorem 6.6) and give a substantial extension (Theorem 5.12). The main open question is: are nearest points dense in the boundary of every closed subset of every reflexive space? Indeed can a proper closed set in a reflexive space fail to have any nearest points? In Section 7 we show that there are some non-Kadec reflexive spaces in which nearest points are dense in the boundary of every closed set.

The differentiability of real functions on normed linear space using generalized subgradients

Abstract The modification of the Clarke generalized subdifferential due to Michel and Penot is a useful tool in determining differentiability properties for certain classes of real functions on a normed linear space. The Gâteaux differentiability of any real function can be deduced from the Gâteaux differentiability of the norm if the function has a directional derivative which attains a constant related to its generalized directional derivative. For any distance function on a space with uniformly Gâteaux differentiable norm, the Clarke and Michel-Penot generalized subdifferentials at points off the set reduce to the same object and this generates a continuity characterization for Gâteaux differentiability. However, on a Banach space with rotund dual, the Frechet differentiability of a distance function implies that it is a convex function. A mean value theorem for the modified generalized subdifferential has implications for Gâteaux differentiability.

MOSCO CONVERGENCE AND THE KADEC PROPERTY

We study the relationship between Wijsman convergence and Mosco convergence for sequences of convex sets. Our central result is that Mosco convergence and Wijsman convergence coincide for sequences of convex sets if and only if the underlying space is reflexive with the dual norm having the Kadec property.

Minimal convex uscos and monotone operators on small sets

We generalize the generic single-valuedness and continuity of monotone operators defined on open subsets of Banach spaces of class (S) and Asplund spaces to monotone operators defined on convex subsets of such spaces which may even fail to have non-support points. This yields differentiability theorems for convex Lipschitzian functions on such sets. From a result about minimal convex uscos which are densely single-valued we obtain generic differentiability results for certain Lipschitzian realvalued functions.

A weak Hadamard smooth renorming of L_1(Omega, mu)

We show that L1(µ) has a weak Hadamard differentiable renorm (i.e. differentiable away from the origin uniformly on all weakly compact sets) if and only if µ is sigma finite. As a consequence several powerful recent differentiability theorems apply to subspaces of L1.

Subdifferentials whose Graphs are not Norm x Weak* Closed

In this note we give examples of convex functions whose subdifferentials have unpleasant properties. Particularly, we exhibit a \pcf on separable Hilbert space such that the graph of its subdifferential is not closed in the product of the norm and bounded weak topologies. We also exhibit a set whose sequential normal cone is not norm closed.