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'rearrangement inequality'
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| Title of object: |
rearrangement inequality |
| Canonical Name: |
RearrangementInequality |
| Type: |
Theorem |
| Created on: |
2001-10-17 01:23:33-04 |
| Modified on: |
2001-11-05 23:40:11-05 |
| Classification: |
msc:26D15 |
| Keywords: |
Inequality, Sequences |
Revision comment (for changes between this and next version):
| Changes for correction #1814 ('rearrangement inequality -- minor correction'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic}
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Content:
Let $x_1,x_2,\ldots,x_n$ and $y_1,y_2,\ldots,y_n$ two sequences of real numbers.
Then the sum
$$x_1y_1+x_2y_2+\cdots+x_ny_n$$
is maximized when the two sequences are ordered in the same way (i.e. $x_1\le x_2\le \cdots \le x_n$ and $y_1\le y_2\le\cdots\le y_n$) and is minimized when the two sequences are ordered in the opposite way (i.e. $x_1\le x_2\le \cdots \le x_n$ and $y_1\ge y_2\ge\cdots\ge y_n$).
\bigskip
This can be seen intuitively as:
If $x_1,x_2,\ldots,x_n$ are the prices of $n$ kinds of items, and $y_1,y_2,\ldots,y_n$ the number of units sold of each, then the highest profit is when you sell more items with high prices and less 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). |
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