Jon D. Vanderwerff

Smooth approximations in Banach spaces

A Banach space that has a locally uniformly convex (LUC) norm whose dual is also LUC is shown to adit C 1 -smooth partitions of unity. It is also established that there is a norm on a Hilbert space with Lipschitz derivative that cannot be approximated uniformly on bounded sets by functions with uniformly continuous second derivative

UNIFORMLY CONVEX FUNCTIONS ON BANACH SPACES

We study the connection between uniformly convex functions f : X -> R bounded above by ||x||^p, and the existence of norms on X with moduli of convexity of power type. In particular, we show that there exists a uniformly convex function f : X -> R bounded above by ||.||2 if and only if X admits a norm with modulus of convexity of power type 2.

Banach Spaces That Admit Support Sets

It is shown that the existence of a closed convex set all of whose points are properly supported in a Banach space is equivalent to the existence of a certain type of uncountable ordered one-sided biorthogonal system. Under the continuum hypothesis, we deduce that this notion is weaker than the existence of an uncountable biorthogonal system.

Epigraphical and Uniform Convergence of Convex Functions

We examine when a sequence of lsc convex functions on a Banach space converges uniformly on bounded sets (resp. compact sets) provided it converges Attouch-Wets (resp. Painleve-Kuratowski). We also obtain related results for pointwise convergence and uniform convergence on weakly compact sets. Some known results concerning the convergence of sequences of linear functionals are shown to also hold for lsc convex functions. For example, a sequence of lsc convex functions converges uniformly on bounded sets to a continuous affine function provided that the convergence is uniform on weakly compact sets and the space does not contain an isomorphic copy of

Constructible convex sets

ell_1

A limiting example for the local ``fuzzy'' sum rule in nonsmooth analysis

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Mazur intersection properties for compact and weakly compact convex sets

We investigate when closed convex sets can be written as countable intersections of closed half-spaces in Banach spaces. It is reasonable to consider this class to comprise the constructible convex sets since such sets are precisely those that can be defined by a countable number of linear inequalities, hence are accessible to techniques of semi-infinite convex programming. We also explore some model theoretic implications. Applications to set convergence are given as limiting examples.

Further arguments for slice convergence in nonreflexive spaces

We show that assuming all the summand functions to be lower semicontinuous is not sufficient to ensure a (strong) fuzzy sum rule for subdifferentials in any infinite dimensional Banach space. From this we deduce that additional assumptions are also needed on functions for chain rules, multiplier rules for constrained minimization problems and Clarke-Ledyaev type mean value inequalities in the infinite dimensional setting.

Biorthogonal Systems in Banach Spaces

Various authors have studied when a Banach space can be renormed so that every weakly compact convex, or less restrictively every compact convex set is an intersection of balls. We first observe that each Banach space can be renormed so that every weakly compact convex set is an intersection of balls, and then we introduce and study properties that are slightly stronger than the preceding two properties respectively. Received by the editors October 8, 1996; revised September 17, 1997. Research partially supported by a Walla Walla College Faculty Development Grant. AMS subject classification: 46B03, 46B20, 46A55. c Canadian Mathematical Society 1998.

On Smooth Variational Principles in Banach Spaces

It is shown that no notion of set convergence at least as strong as Wijsman convergence but not as strong as slice convergence can be preserved in superspaces. We also show that such intermediate notions of convergence do not always admit representations analogous to those given by Attouch and Beer for slice convergence, and provide a valid reformulation. Some connections between bornologies and the relationships between certain gap convergences for nonconvex sets are also observed.