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Triangle Difference Inequality

A useful variation on the triangle inequality is that the length of any side of a triangle is greater than the absolute difference of the lengths of the other two sides:

$\displaystyle \zbox {\Vert\underline{u}-\underline{v}\Vert \geq \left\vert\Vert\underline{u}\Vert - \Vert\underline{v}\Vert\right\vert}
$



Proof: By the triangle inequality,

\begin{eqnarray*}
\Vert\underline{v}+ (\underline{u}-\underline{v})\Vert &\leq & \Vert\underline{v}\Vert + \Vert\underline{u}-\underline{v}\Vert \\
\,\,\Rightarrow\,\,\Vert\underline{u}-\underline{v}\Vert &\geq& \Vert\underline{u}\Vert - \Vert\underline{v}\Vert.
\end{eqnarray*}

Interchanging $ \underline{u}$ and $ \underline{v}$ establishes the absolute value on the right-hand side.


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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