%%% This file is part of PlanetMath snapshot of 2009-01-12
%%% Primary Title: city-block metric
%%% Primary Category Code: 54E35
%%% Filename: CityBlockMetric.tex
%%% Version: 5
%%% Owner: akrowne
%%% Author(s): akrowne
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\begin{document}

 The \emph{city-block metric}, defined on $\mathbb{R}^n$, is

$$ d(a,b) = \sum_{i=1}^n |b_i-a_i| $$

where $a$ and $b$ are \htmladdnormallink{vectors}{http://planetmath.org/encyclopedia/Vector2.html} in $\mathbb{R}^n$ with $a=(a_1,\ldots,a_n)$ and $b=(b_1,\ldots,b_n)$.

In two \htmladdnormallink{dimensions}{http://planetmath.org/encyclopedia/Dimension3.html} and with discrete-valued vectors, when we can picture the set of \htmladdnormallink{points}{http://planetmath.org/encyclopedia/Point.html} in $\mathbb{Z} \times \mathbb{Z}$ as a \htmladdnormallink{grid}{http://planetmath.org/encyclopedia/LatticeInMathbbRn.html}, this is simply the \htmladdnormallink{number}{http://planetmath.org/encyclopedia/Number.html} of \htmladdnormallink{edges}{http://planetmath.org/encyclopedia/Adjacent.html} between points that must be traversed to get from $a$ to $b$ within the grid. This is the same problem as getting from corner $a$ to $b$ in a rectilinear downtown \htmladdnormallink{area}{http://planetmath.org/encyclopedia/SurfaceArea.html}, hence the name ``city-block metric.''

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