city-block metric

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

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

where $a$ and $b$ are vectors in ${ℝ}^n$ 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.”