Geometric Mean Value Theorems for the Dini Derivative

A new class of mean value theorems, which involve geometry of the function domain, is introduced. Roughly speaking, if f maps a Banach space (X, ||·||) into R ∪ {+∞} and a, b ∈ X are such that f(a) > f(b), then there is a point x ∈ B (a, ||a − b||) at which the Dini derivative df(x*; h) is nonnegative for every direction h from some cone. Examples of applications are given which show an advantage of such results over standard mean value theorems.