I am studying algorithms and datastructures, and in CLRS chapter 33.4, the exercise 33.4-4 states the following:

*We can define the distance between two points in ways other than euclidean. In the plan, the $ L_m$ distance between points $ p_1$ and $ p_2$ are given by the expression $ (|x_1-x_2|^m+|y_1-y_2|^m)^{1/m}$ . euclidean distance, there, is $ L_2$ -distance. Modify the closest pair algorithm to use the $ L_1$ -distance, which is also known as the Manhattan distance.*

I understand its an easy modification, however when I’ve been comparing my intuition with that of solutions provided online (e.g. https://sites.math.rutgers.edu/~ajl213/CLRS/CLRS.html), all I can find is that they all point to the number of points in the $ \delta$ x $ 2\delta$ is increased to 10 (from 8), adding one point to the middle of both squares formed by the rectangle (*here marked with red below*):

Can anyone explain to my why this is the case? Intuitively I cannot see why it should be the case that the number of points in the rectange should increase, when switching to another **$ L_m$ -distance**?

The **Manhatten distance** $ \geq$ **Euclidean distance** from what I can see in all instances, but why should this infer, an increase in points found in the $ \delta$ x $ 2\delta$ space.

What am I missing?