In mathematics, the Kantorovich inequality is a particular case of the Cauchy–Schwarz inequality, which is itself a generalization of the triangle 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. (See vector space, inner product, and normed 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
≥
0
,
0
<
a
≤
x
i
≤
b
for
i
=
1
,
…
,
n
.
{\displaystyle p_{i}\geq 0,\quad 0