# Kantorovich inequality

In

mathematics , the**Kantorovich inequality**is a particular case of theCauchy-Schwarz inequality , which is itself a generalization of thetriangle inequality .The triangle inequality states that the length of two sides of any triangle, added together, will be equal to or greater than the length of the third side. In simplest terms, the Kantorovich inequality translates the basic idea of the triangle inequality into the terms and notational conventions of

linear programming . (Seevector space ,inner product , andnormed vector space for other examples of how the basic ideas inherent in the triangle inequality--line segment and distance--can be generalized into a broader context.)More formally, the Kantorovich inequality can be expressed this way:

:Let $p\_i\; geq\; 0,$ $0leq\; b\; math>\; for$ i=1,\; dots\; ,n.$:Let$ A\_n=\{1,2,dots\; ,n\}.$:Then\; :::$ left\; (\; sum\_\{i=1\}^n\; p\_ix\_i\; ight\; )\; left\; (sum\_\{i=1\}^n\; frac\{p\_i\}\{x\_i\}\; ight\; )$$

::$leq\; frac\{(a+b)^2\}\{4ab\}\; left\; (sum\_\{i=1\}^n\; p\_i\; ight\; )^2-frac\{(a-b)^2\}\{4ab\}\; cdot\; min\; left\{\; left\; (sum\_\{i\; in\; X\}p\_i-sum\_\{j\; in\; Y\}p\_j\; ight\; )^2,:,\; \{X\; cup\; Y=A\_n\},\{X\; cap\; Y=varnothing\}\; ight\}.$

The Kantorovich inequality is used in

convergence analysis ; it bounds the convergence rate of Cauchy'ssteepest descent .Equivalents of the Kantorovich inequality have arisen in a number of different fields. For instance, the

Bunyakovsky inequality , theWielandt inequality , and theCauchy-Schwarz inequality are equivalent to the Kantorovich inequality and all of these are, in turn, special cases of theHölder inequality .The Kantorovich inequality is named after Soviet economist, mathematician, and

Nobel Prize winnerLeonid Kantorovich , a pioneer in the field oflinear programming .**References*** [

*http://mathworld.wolfram.com/KantorovichInequality.html Eric W. Weisstein. "Kantorovich Inequality." From MathWorld--A Wolfram Web Resource.*]* [

*http://planetmath.org/encyclopedia/KantorovichInequality.html Planet Math entry on "Kantorovich inequality"*]* [

*http://carbon.cudenver.edu/~hgreenbe/glossary/index.php?page=K.html Mathematical Programming Glossary entry on "Kantorovich inequality"*]**External links*** [

*http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Kantorovich.html Biography of Leonid Vitalyevich Kantorovich*]

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