Suppose you expanded every point of a coastline into a small disk (i.e. take the minkowski sum [1] of the coastline and a disk). Is the perimeter of the resulti

Suppose you expanded every point of a coastline into a small disk (i.e. take the minkowski sum [1] of the coastline and a disk). Is the perimeter of the resulting shape still infinite? The coastline had infinite perimeter because as you zoomed it got jankier and jankier. But once the janks get smaller than the disk, they start being hidden by disks from the surrounding points. You can't jump in and out anymore on tiny scales, because you run into the adjacent disks. This smooths out the noise and things converge instead of diverging. ... I think. A river's length would smooth out for essentially the same reason. The set of points equidistant from both sides of the river is smoothed out compared to the sides because points on the side that are closer to the center of the river will hide jank from the nearby further points. That's my intuition anyways. Not sure how it actually plays out. Edit And if you go by "shortest path within the river from source to sink" as the length then it's definitely finite. 1: https://en.wikipedia.org/wiki/Minkowski_addition https://en.wikipedia.org/wiki/Minkowski_addition