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,

[math]\displaystyle{ \mathrm{d} f(x) (y - x) \leq f(y) - f(x); }[/math]

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

[math]\displaystyle{ \big( \mathrm{d} f(x) - \mathrm{d} f(y) \big) (x - y) \geq 0. }[/math]

References

• Kachurovskii, R. I. (1960). "On monotone operators and convex functionals". Uspekhi Mat. Nauk 15 (4): 213–215. 
• Showalter, Ralph E. (1997). Monotone operators in Banach space and nonlinear partial differential equations. Mathematical Surveys and Monographs 49. Providence, RI: American Mathematical Society. pp. 80. ISBN 0-8218-0500-2. https://archive.org/details/monotoneoperatio00show.  MR1422252 (Proposition 7.4)