Rearrangement Inequality

$x\_1\; \backslash leq\; \backslash cdots\; \backslash leq\; x\_n\backslash quad\; \backslash mbox\{and\}\backslash quad\; y\_1\; \backslash leq\; \backslash cdots\; \backslash leq\; y\_n$

$x\_\{\backslash sigma\; (1)\},\; \backslash dots\; ,x\_\{\backslash sigma\; (n)\}$

$x\_1y\_1\; +\; \backslash cdots\; +\; x\_ny\_n\; \backslash geq\; x\_\{\backslash sigma\; (1)\}y\_1\; +\; \backslash cdots\; +\; x\_\{\backslash sigma\; (n)\}y\_n\; \backslash geq\; x\_ny\_1\; +\; \backslash cdots\; +\; x\_1y\_n.$

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