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,

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

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

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

References

• Kachurovskii, I. R. (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. MR1422252 (Proposition 7.4)