Geometric mean value theorems for the Dini derivative (Q1892560)

The authors introduce a new class of mean value problems, which involve geometry of the function domain. Roughly speaking, if \(f\) maps a Banach space \((X, |\cdot |)\) into \(\mathbb{R}\cup \{+\infty\}\) and \(a,b\in X\) are such that \(f(a)> f(b)\), then there is a point \(x\in B (a,|a- b|)\) at which the Dini derivative \({\mathbf d} f(x^*; h)\) is nonnegative for every direction \(h\) from some cone. The advantage of such results over standard mean value theorems is shown by examples.