# Shortest maximum pairwise distance of points in a circle of radius R

Given a positive real number $$R$$ and $$n$$ fixed points in a plane, find the largest possible value $$M$$ such that if the pairwise distance between the $$n$$ points is less or equal to $$M$$, there exists a circle of radius $$R$$ that contains all $$n$$ points.

I realize this is related to the smallest enclosing circle problem, but it is not the same. Basically, I have a set of points, and I know all the pairwise distances between them, and I am given a fixed radius. I want to be able to say that if the maximum pairwise distance between these points is less than $$M$$, then there is definitely a circle of radius $$R$$ that encloses all of them, where this M is as large as possible.

Obviously, for two points, the answer is $$2R$$. For three points, it gets more complicated, but using some geometry, I believe the answer is $$\sqrt{2}R$$ (as in $$\sqrt{R^2 + R^2}$$), but I haven't been able to prove it. I would like to generalize this to $$n$$ points, where I am fine with the answer being dependent on $$n$$.

My hypothesis is that $$\sqrt{2}R \leq M \leq 2R$$ for all $$n$$, but I would be mainly interested in how large $$M$$ can be made as the number of points increases.

I don't have a complete solution but I can prove that your conclusion for $$n=3$$ points is wrong. Suppose that the maximum pairwise distance between 3 points is $$M$$. WLOG, suppose that the maximum distance $$M$$ is reached between points $$A$$ and $$B$$.
Where is the third point $$C$$? It has to be somewhere in the pink region. The borders of that region are segment $$AB$$ and two arcs, each encompassing $$60^\circ$$ with centers at points $$A$$ and $$B$$. Wherever you put the third point ($$C',C''$$), you can always cover the whole triangle with the blue circle. But the blue circle cannot be any smaller because in that case it could not cover the triangle $$ABC$$. And in that case:
$$M=R\sqrt{3}$$
• Thanks! Just realized my flaw for the $n=3$ case. Theoretically, since points can be arbitrarily close together, I think $M$ cannot be any larger than $R\sqrt{3}$, and since adding more points only puts more restrictions on where the next one can go, I don't think $M$ gets any smaller, either. For my practical application, though, it would be interesting to see if $M$ could be made larger with a relatively high probability for points that are reasonably "spread out." – Dennis Mar 27 '19 at 21:07