Greatest minimum distance of $9$ points in a unit square

Question

Put $9$ points in a unit square such that the distance between any $2$ points is less or equal to $1$. What is the greatest value of the minimum distance among these $9$ points?

Can it be $\dfrac{\sqrt2}3$? Thanks.

Answer 1

See https://en.wikipedia.org/wiki/Circle_packing for some results which may be of relevance.

Upper bound

If we ignore the restriction on the separation being no more then $1$ then the greatest minimum separation is $\dfrac{1}{2}$. In this case, two of the separations are as large as $\sqrt 2$ and this suggests that the greatest minimum separation for your problem will be significantly less than $\dfrac{1}{2}$ and therefore less than $\dfrac{\sqrt2}3$.

Lower bound

The convex hull of the nine points has to have diameter less than $1$. Considering the case that this region is a circle, the greatest minimum separation for your problem is $\sin\dfrac{\pi}{8}\approx 0.383$. This is obtained by putting one point in the centre with the other $8$ points equally spaced on a circle of radius $\dfrac{1}{2}$.