Upper bound on the minimum distance between $N$ points chosen inside the unit circle?

Question

I guess this is a well-known problem but I'm not sure where to find it on the web.

$N \ge 2$ points are chosen in the interior or the boundary of the unit circle. What is the best upper bound on the minimum distance between two of these points?

Given a configuration of $N$ such points, let's call the minimum distance $d_{\min}(N)$.We seek $\max\{d_{\min}(N)\}$. Some examples for small $N$:

$\max\{d_{\min}(2)\}=2$ (diameter)

$\max\{d_{\min}(3)\} \ge \sqrt{3}$ (equilateral triangle)

$\max\{d_{\min}(4)\} \ge \sqrt{2}$ (square)

$\max\{d_{\min}(5)\} \ge 2\sin(\pi/5)$ (regular pentagon)

$\max\{d_{\min}(6)\} \ge 1$ (regular hexagon)

$\max\{d_{\min}(7)\} \ge 1$ (regular hexagon plus the center)

This shows the answer is not $2\sin(\pi/N)$, which you would get from distributing the points equally along the circumference (the pattern breaks for $N=7$).

Answer 1

Let's sketch the equivalence of the problem above:

1. Given $N \ge 2$, find $N$ points in the closed unit disk such that the minimum distance $d$ between any pair of points is maximized.

and the "circle packing in a circle" problem:

2. Given $N \ge 2$, place $N$ circles of largest possible equal radius $r$ inside the unit disk so that their interiors have pairwise empty intersections.

A solution of the second problem gives us the $N$ centers of the circles all within a disk of radius $1-r$ and a minimum distance between any two centers of $2r$. Dilating these points to locations inside a unit disk gives:

$$ d = \frac{2r}{1-r} $$

Conversely, given a solution of the first problem, any pair of the $N$ points are not closer than $d$, so circles around these points of radius $d/2$ will not have overlapping interiors and will be contained in a disk of radius $1 + (d/2)$. Contracting the containing disk to radius one provides us with $N$ circles packed in the unit disk having equal radius:

$$ r = \frac{d/2}{1 + (d/2)} $$

The Reader is kindly asked to verify that composition of these rational expressions gives an identity, so that it suffices to check monotonicity of either of them. For instance we could rewrite the latter expression:

$$ r = \frac{1}{(2/d) + 1} $$

This makes it evident that as $d$ increases, $r$ increases. It follows that extremal solutions to the first problem correspond with extremal solutions to the second problem (and conversely).

A third problem, whose equivalence with the second one should be clear:

3. Given $N \ge 2$, place $N$ circles of unit radius inside a bounding circle of smallest possible radius $R$ so that the unit circles' interiors do not overlap.

As @achille hiu notes, packomania is a good source for the best known circles-packed-in-a-circle arrangements. According to the Wikipedia article linked above, the smallest case for which optimality of such arrangements is conjectural (unproven) is $N=14$. The optimality proof for $N=13$ was published by F. Fodor (2003), "The Densest Packing of 13 Congruent Circles in a Circle", Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry) 44:2, pp. 431–440.