# city-block metric

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 $a=({a}_{1},\mathrm{\dots},{a}_{n})$ and $b=({b}_{1},\mathrm{\dots},{b}_{n})$.

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.”

Title | city-block metric |
---|---|

Canonical name | CityblockMetric |

Date of creation | 2013-03-22 12:12:57 |

Last modified on | 2013-03-22 12:12:57 |

Owner | akrowne (2) |

Last modified by | akrowne (2) |

Numerical id | 9 |

Author | akrowne (2) |

Entry type | Definition |

Classification | msc 54E35 |

Synonym | city-block distance |

Synonym | taxicab metric |