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(x) : V → R at each point x in K. (In fact, df(x) is an element of the continuous dual space V∗.) Then the following are equivalent: f is a convex function; for all x and y in K, d f ( x ) ( y − x ) ≤ f ( y ) − f ( x ) ; {\displaystyle \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, ( d f ( x ) − d f ( y ) ) ( x − y ) ≥ 0.