Mean value theorems and convexity: an example of cross-fertilization of two mathematical items (Q827538)

In this interesting paper, the author shows on particular examples how mean value theorems of differential calculus and results of convexity theory can have impact on each other. First, he considers the real-value case and then the vectorial case is studied. A new result in the first part is related to the mean-value points set in the Lagrange mean value theorem, by showing that the fact that this set is an interval has equivalent formulations related to the level sets of derivative function, to the monotonicity of derivative function or to the convexity or concavity of the function.