Applied Analysis Seminar

Monday January 28 and Monday February 4, in K9509, SFU

3:30 - 4:20
Rafal Goebel

Title: "Cone-monotone functions in Banach spaces - examples of lack of regularity."

Abstract: The simplest examples of cone monotone functions - functions for which f(y) >= f(x)$ whenever y-x is an element of a given cone - are provided by nondecreasing functions on the real line. These have several immediate regularity properties, the most intuitive of which may be the at most countable number of discontinuities. More generally, coordinate-wise nondecreasing functions in finite dimensions - equivalently, functions monotone with respect to the nonnegative cone - are measurable, almost everywhere continuous, and almost everywhere Frechet differentiable.

Recently, Borwein, Burke, and Lewis showed that functions on a separable Banach space, monotone with respect to a convex cone with nonempty interior, are differentiable except at points of an appropriately understood null set. The goal of the talk is to briefly survey the known facts on cone monotone functions in finite dimensions and demonstrate, through examples, how these facts, and the results of Borwein et al., do not extend to a general Banach space.