Kachurovskii's theorem

In mathematics, Kachurovskii's theorem is a theorem relating the convexity of a function on a Banach space to the monotonicity of its Fréchet derivative.

Statement of the theorem

Let K be a convex subset of a Banach space V and let f : K → R ∪ be an extended real-valued function that is Fréchet differentiable with derivative df : V → R at each point x in K. Then the following are equivalent:f is a convex function;

for all x and y in K,
df is an monotone operator, i.e., for all x and y in K,