Kachurovskii's theorem

Template:Short description 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,

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

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

$(\mathrm{d}f(x)-\mathrm{d}f(y))(x-y)\ge 0.$

References

• Template:Cite journal
• Template:Cite book Template:MathSciNet (Proposition 7.4)