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642:{\displaystyle {\begin{aligned}&{}\qquad \left(\sum _{i=1}^{n}p_{i}x_{i}\right)\left(\sum _{i=1}^{n}{\frac {p_{i}}{x_{i}}}\right)\\&\leq {\frac {(a+b)^{2}}{4ab}}\left(\sum _{i=1}^{n}p_{i}\right)^{2}-{\frac {(a-b)^{2}}{4ab}}\cdot \min \left\{\left(\sum _{i\in X}p_{i}-\sum _{j\in Y}p_{j}\right)^{2}\,:\,{X\cup Y=A_{n}},{X\cap Y=\varnothing }\right\}.\end{aligned}}}
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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
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There is also Matrix version of the
Kantorovich inequality due to Marshall and Olkin (1990). Its extensions and their applications to statistics are available; see e.g. Liu and Neudecker (1999) and Liu et al. (2022).
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Liu, Shuangzhe, Leiva, VĂctor, Zhuang, Dan, Ma, Tiefeng and
Figueroa-Zúñiga, Jorge I., Matrix differential calculus with applications in the multivariate linear model and its diagnostics.
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for other examples of how the basic ideas inherent in the triangle inequality—line segment and distance—can be generalized into a broader context.)
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Liu, Shuangzhe and
Neudecker, Heinz, A survey of Cauchy-Schwarz and Kantorovich-type matrix inequalities. Statistical Papers 40 (1999) 55-73.
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Equivalents of the
Kantorovich inequality have arisen in a number of different fields. For instance, the
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are equivalent to the
Kantorovich inequality and all of these are, in turn, special cases of the
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Marshall, A. W. and Olkin, I., Matrix versions of the Cauchy and
Kantorovich inequalities.
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152:{\displaystyle p_{i}\geq 0,\quad 0<a\leq x_{i}\leq b{\text{ for }}i=1,\dots ,n.}
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The
Kantorovich inequality is named after Soviet economist, mathematician, and
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More formally, the
Kantorovich inequality can be expressed this way:
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Mathematical
Programming Glossary entry on "Kantorovich inequality"
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667:Cauchy–Schwarz–Bunyakovsky inequality
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