Kachurovskii's theorem

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) \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.

References

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