Template:AotD

Rearrangement Inequality

The Rearrangement Inequality states that, if $A=\{a_1,a_2,\cdots,a_n\}$ is a permutation of a finite set (in fact, multiset) of real numbers and $B=\{b_1,b_2,\cdots,b_n\}$ is a permutation of another finite set of real numbers, the quantity $a_1b_1+a_2b_2+\cdots+a_nb_n$ is maximized when ${A}$ and ${B}$ are similarly sorted (that is, if $a_k$ is greater than or equal to exactly ${i}$ of the other members of $A$ , then ${b_k}$ is also greater than or equal to exactly ${i}$ of the other members of $B$ ). Conversely, $a_1b_1+a_2b_2+\cdots+a_nb_n$ is minimized when $A$ and $B$ are oppositely sorted (that is, if $a_k$ is less than or equal

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Template:AotD

Rearrangement Inequality