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− | ===[[ | + | ===[[Rearrangement Inequality]]=== |
− | ''' | + | The '''Rearrangement Inequality''' states that, if <math>A=\{a_1,a_2,\cdots,a_n\}</math> is a [[permutation]] of a [[finite]] [[set]] (in fact, [[multiset]]) of [[real number]]s and <math>B=\{b_1,b_2,\cdots,b_n\}</math> is a permutation of another finite set of real numbers, the quantity <math>a_1b_1+a_2b_2+\cdots+a_nb_n</math> is maximized when <math>{A}</math> and <math>{B} </math> are similarly sorted (that is, if <math>a_k</math> is greater than or equal to exactly <math>{i}</math> of the other members of <math>A</math>, then <math> {b_k} </math> is also greater than or equal to exactly <math>{i}</math> of the other members of <math>B</math>). Conversely, <math>a_1b_1+a_2b_2+\cdots+a_nb_n</math> is minimized when <math>A</math> and <math>B</math> are oppositely sorted (that is, if <math>a_k</math> is less than or equal |
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Revision as of 23:05, 13 December 2007
Rearrangement Inequality
The Rearrangement Inequality states that, if
is a permutation of a finite set (in fact, multiset) of real numbers and
is a permutation of another finite set of real numbers, the quantity
is maximized when
and
are similarly sorted (that is, if
is greater than or equal to exactly
of the other members of
, then
is also greater than or equal to exactly
of the other members of
). Conversely,
is minimized when
and
are oppositely sorted (that is, if
is less than or equal