Thanks for the clarifications! Silly me, my mistake was taking "infinitesimal" too literally. OK, so one way to visualize this is finding short paths in the plane that must avoid "obstructions". Specifically here the obstructions are an infinite grid of (open?) square blocks of width 1-2h centered at the half-integers, with sides parallel to the axis. It's interesting to see how the bundles of minimal paths changes as h varies from 1/2 (taxicab) to 0 (Euclidean). We can also play with the design of the city. For example we could have a hexagonal tiling, or change the blocks to be an "orchard" of circular obstructions, or little linear cuts you can't cross, or even fractals. And instead of an endless grid you could have just a single forbidden disk, say the unit circle. Then almost all minimal paths are Euclidean, except a subset where the endpoints are "shadowed" from each other by the obstruction. Maybe roboticists that do path planning and obstacle avoidance might have some ideas?