Why is this proof that the distance between two of any five points in a square $\geq \frac{\sqrt{2}}{2}$ wrong?

Question

Why is this proof that the distance between two of any five points in a square $\geq \frac{\sqrt{2}}{2}$ wrong?
This is (again) from Zeitz. The proof is somewhat like this: let us place four of the five points at the corners for maximal separation. Now the furthest the point can be from them is at the center, so the distance is $\frac{\sqrt{2}}{2}$.

What's the mistake?

Answer 1

If we only place four points in the square, it is clear that placing the points at the corners maximizes the smallest distance between those four points. However, it is not clear that starting with four points at the corners and adding a point will maximize the smallest distance between five points.

Consider a slightly different problem. "Can we place $3$ points in a disk of radius $1$ such that the distance between any pair of points is at least $\frac{3}{2}$?" Let's use your argument:

Let us place two of the three points at endpoints of a diameter of the disk for maximal separation. Now the furthest the point can be from them is on the midpoint of the arc between the first two points, so the distance is $\sqrt{2} < \frac{3}{2}$.

However, it is possible to place $3$ points on the circle such that the distance between any pair of points is $\sqrt{3} > \frac{3}{2}$. This can be achieved by picking $3$ points on the boundary of the disk that are vertices of an equilateral triangle. As you can see, the "greedy" approach of maximizing the distance between the first two points and then adding a third point did not work. The "greedy" approach did work for your problem, but it doesn't work in general, which is why it isn't a valid argument.