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city-block metric (Definition)

The 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 vectors in $\mathbb{R}^n$ with $a=(a_1,\ldots,a_n)$ and $b=(b_1,\ldots,b_n)$

In two dimensions and with discrete-valued vectors, when we can picture the set of points in $\mathbb{Z} \times \mathbb{Z}$ as a grid, this is simply the number of edges 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 area, hence the name ``city-block metric.''




"city-block metric" is owned by akrowne.
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Other names:  city-block distance, taxicab metric
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Cross-references: area, edges, number, grid, points, dimensions, vectors
There are 4 references to this entry.

This is version 5 of city-block metric, born on 2002-01-23, modified 2003-02-23.
Object id is 1552, canonical name is CityBlockMetric.
Accessed 20612 times total.

Classification:
AMS MSC54E35 (General topology :: Spaces with richer structures :: Metric spaces, metrizability)

Pending Errata and Addenda
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