Triangle Difference Inequality

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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,

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

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

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Mathematics of the DFT
    Geometric Signal Theory
       The Inner Product
          Triangle Difference Inequality