Kachurovskii's theorem

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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.

[edit] 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.

[edit] References

  • Showalter, Ralph E. (1997). Monotone operators in Banach space and nonlinear partial differential equations, Mathematical Surveys and Monographs 49. Providence, RI: American Mathematical Society, 80. ISBN 0-8218-0500-2.  MR1422252 (Proposition 7.4)