mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Kantorovich inequality is a particular case of the

Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality fo ...

, which is itself a generalization of the

triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...

. 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 Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, li ...

. (See

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...

, and

normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...

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,\quad 0 < a \leq x_i \leq b\texti=1, \dots ,n.

:Let

A_n=\.

:Then ::

\begin
&  \qquad \left( \sum_^n p_ix_i \right ) \left (\sum_^n \frac \right) \\
& \leq \frac \left (\sum_^n p_i \right )^2
-\frac \cdot \min \left\.
\end

The Kantorovich inequality is used in convergence analysis; it bounds the convergence rate of Cauchy's

steepest descent In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the ...

. Equivalents of the Kantorovich inequality have arisen in a number of different fields. For instance, the Cauchy–Schwarz–Bunyakovsky inequality and the Wielandt inequality are equivalent to the Kantorovich inequality and all of these are, in turn, special cases of the

Hölder inequality Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality * Hölder mean * Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a modul ...

. The Kantorovich inequality is named after Soviet economist, mathematician, and

Nobel Prize The Nobel Prizes ( ; sv, Nobelpriset ; no, Nobelprisen ) are five separate prizes that, according to Alfred Nobel's will of 1895, are awarded to "those who, during the preceding year, have conferred the greatest benefit to humankind." Alfr ...

winner

Leonid Kantorovich Leonid Vitalyevich Kantorovich ( rus, Леони́д Вита́льевич Канторо́вич, , p=lʲɪɐˈnʲit vʲɪˈtalʲjɪvʲɪtɕ kəntɐˈrovʲɪtɕ, a=Ru-Leonid_Vitaliyevich_Kantorovich.ogg; 19 January 19127 April 1986) was a Soviet ...

, a pioneer in the field of

. 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).

References

* * {{PlanetMath, urlname=KantorovichInequality, title=Cauchy-Schwarz inequality
Mathematical Programming Glossary entry on "Kantorovich inequality"
* Marshall, A. W. and Olkin, I., Matrix versions of the Cauchy and Kantorovich inequalities.

Aequationes Mathematicae ''Aequationes Mathematicae'' is a mathematical journal. It is primarily devoted to functional equations, but also publishes papers in dynamical systems, combinatorics, and geometry. As well as publishing regular journal submissions on these topics ...

40 (1990) 89–93. * Liu, Shuangzhe and Neudecker, Heinz, A survey of Cauchy-Schwarz and Kantorovich-type matrix inequalities. Statistical Papers 40 (1999) 55-73. * Liu, Shuangzhe, Leiva, Víctor, Zhuang, Dan, Ma, Tiefeng and Figueroa-Zúñiga, Jorge I., Matrix differential calculus with applications in the multivariate linear model and its diagnostics.

Journal of Multivariate Analysis The ''Journal of Multivariate Analysis'' is a monthly peer-reviewed scientific journal that covers applications and research in the field of multivariate statistical analysis. The journal's scope includes theoretical results as well as applicat ...

188 (2022) 104849. https://doi.org/10.1016/j.jmva.2021.104849

External links

Biography of Leonid Vitalyevich Kantorovich
Theorems in analysis Inequalities