Natural example where $\ell_\infty$ distance appears.

Question

The $\ell_2$ distance has a natural connotation: the straight line distance between two points "as the crow flies".

Similarly, the $\ell_1$ distance has a natural connotation: the length of a path between two points in a grid where you're only allowed to walk along grid lines. Or more simply, the length of a path between two locations in Manhattan.

Is there a similar natural connotation for the $\ell_\infty$ metric ?

Answer 1

Isn't $\ell_{\infty}$ just "Manhattan-with-diagonals", i.e. something like the "Washington, DC" metric? Instead of the distance on the grid graph, it's just the distance on the grid graph when diagonals are added, i.e. every node has degree 8 instead of 4 (in 2 dimensions). With the same intuition, the $\ell_{\infty}$ distance between two points is the number of moves a King in chess would take to get between them.

Answer 2

In more common language, the $l_\infty$ distance is (proportional to) how long it takes to get from one point to the other when you can travel in multiple dimensions at once, each dimension at the same speed.