rearrangement inequality

Let x1,x2,,xn and y1,y2,,yn two sequences of positive real numbers. Then the sum


is maximized when the two sequences are ordered in the same way (i.e. x1x2xn and y1y2yn) and is minimized when the two sequences are ordered in the opposite way (i.e. x1x2xn and y1y2yn).

This can be seen intuitively as: If x1,x2,,xn are the prices of n kinds of items, and y1,y2,,yn the number of units sold of each, then the highest profit is when you sell more items with high prices and fewer items with low prices (same ordering), and the lowest profit happens when you sell more items with lower prices and less items with high prices (opposite orders).

Title rearrangement inequality
Canonical name RearrangementInequality
Date of creation 2013-03-22 11:47:32
Last modified on 2013-03-22 11:47:32
Owner drini (3)
Last modified by drini (3)
Numerical id 9
Author drini (3)
Entry type Theorem
Classification msc 26D15
Classification msc 20L05
Classification msc 22A22
Related topic ChebyshevsInequality