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

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.”