If we think of the l1 distance as a grid-distance between points, then we can think of l2 distance as what we get when we "shortcut" the grid by going "inside" a cell. 


Making the grid finer doesn't change the l1 distance, so there's no obvious sense in which the l2 distance can be seen as a limiting version of the l1. 


So here's my question: is there any way to take an l1-like distance and extract an l2 like distance from it (possibly as a limiting case). I'm asking because i have a distance defined in a discrete space that has an l1-like behaviour, and I'd like to generalize it as the discrete space gets finer and finer, but I want to end up with a distance that goes "directly" through the space like l2. 


apologies if this is way too vague. 



Making an l_2 distance out of l_1 distance
L\_2 can be viewed as the limit of a sequence of metrics whose metric balls are (in the plane) regular polygons with 2n sides, where n goes to infinity. In higher dimensions one doesn't have enough regular polytopes to make a limit out of them, but it works just as well to use irregular ones. This technique is sometimes useful in computational geometry, because the polygonal metrics allow for faster range searching data structures than the Euclidean one.


If you want a distance in some sort of discrete graph (the way that L\_1 can be viewed as distance in a regular grid) but that approaches planar L\_2 in the limit, try using the pinwheel tiling. If you measure the shortest path along tile edges between any two points on the edges of any two tiles, and then repeatedly subdivide to produce a finer pinwheel tiling, the shortest path distance will approach the Euclidean distance in the limit.


As for Kore min's suggestion of looking into metric embedding: any metric can be embedded into L\_infinity with no distortion at all. And in the plane, L\_1 is the same as L\_infinity (though they are quite different in higher dimensions) so it might have something to do with what you're talking about. See e.g. [this Wikipedia article](http://en.wikipedia.org/wiki/Tight_span).


