GEOMETRIC MEAN-VALUE THEOREMS FOR THE DINI DERIVATIVE

A new class of mean value theorems, which involve geometry of the func tion domain, is introduced. Roughly speaking, if f maps a Banach space (X, parallel to .parallel to) into R boolean OR {+infinity} and a, b is an element of X are such that f(a) > f(b), then there is a point x is an element of B (a, parallel to a - b parallel to) at which the Din i derivative d f(x; h) is nonnegative for every direction h from some cone. Examples of applications are given which show an advantage of s uch results over standard mean value theorems. (C) 1995 Academic Press , Inc.