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 &rarr; R &cup; {+&infin;} be an extended real-valued function that is Fréchet differentiable with derivative df(x) : V &rarr; R at each point x in K. (In fact, df(x) is an element of the continuous dual space V&lowast;.) Then the following are equivalent:


 * f is a convex function;
 * for all x and y in K,


 * $$\mathrm{d} f(x) (y - x) \leq f(y) - f(x);$$


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


 * $$\big( \mathrm{d} f(x) - \mathrm{d} f(y) \big) (x - y) \geq 0.$$