incorrect comment on Advent of Code 2018 Day 6 Part 1

[SabriAl-Safi]

The notebook states:

Originally, I thought that if a point on the perimeter of the [bounding] box was closest to point p, then p has an infinite area. That's true for p that are in the corners, but need not be true for all p (consider a set of 5 points, 4 in the corners of a box, and one near the center; the center point will have a diamond-shaped area, and unless it is at the exact center, it will "leak" outside one of the edges). So I expand the bounding box by a margin.

I tried to replicate this with a drawing of the described situation, but found that the area of the point near the center was still infinite:

I believe that you don't need a buffer on the perimeter. In the Manhatten distance metric, traveling "across then up" is the equal-best way to get between two points. Thus if the closest point to A on the perimeter is B, and B is itself closest to (e), then A is closest to (e).

This fails in the normal Euclidean plane because if you move far enough to the right you will eventually be closer to (b) then to (e) (imagine a circle through (e) with center very far to the right - the circumference through (e) locally appears to be a straight vertical line).