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Consider a collection $C_1,C_2, \dots, C_n$ of circles in the plane and suppose that the center of $C_i$ is $o_i$ and the radius of $C_i$ is $r_i$. We will define the relative distance between the circles $C_i$ and $C_j$ as $$\frac {d(o_1,o_2)}{r_1+r_2},$$ where $d(x,y)$ is the Euclidean distance. So the relative distance is $\le 1$ if the two circles are intersecting, and $\ge 1$ if they are non overlapping.

What is the smallest possible number of distinct relative distances?

For unit circles this is Erdos' distinct distances problem that was famously settled by Guth and Katz.

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  • $\begingroup$ Are you ranging over choices of $r_i$ or are the $r_i$ part of the input to the question? $\endgroup$
    – Sam Hopkins
    Commented Feb 6, 2020 at 23:20
  • $\begingroup$ Dear @SamHopkins, The "smallest possible number" is a function of $n$ and the input is an arbitrary collections of $n$ circuits. (Each circuit is determined by its center and radius.) $\endgroup$
    – Gil Kalai
    Commented Feb 7, 2020 at 8:14
  • $\begingroup$ I am not sure that the relative distance is the most appropriate measure to take. Another option would be to take the cross ratio between the four points $o_i, x_{ij}, y_{ij},o_j$ where $x_{ij}$ and $y_{ij}$ are the intersections of the line segment $[o_i,o_j]$ with $C_i$ and $C_j$ respectively. $\endgroup$
    – Gil Kalai
    Commented Feb 9, 2020 at 6:33
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    $\begingroup$ Do you have an example when it can be less than for points? oeis.org/A186704 $\endgroup$
    – domotorp
    Commented Feb 19, 2020 at 21:55
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    $\begingroup$ @domotorp, Oops sorry we do have 4 (pairwise tangent) circles in the plane of relative distance 1. I dont know if other relative distances allow more. So it is less than for points. $\endgroup$
    – Gil Kalai
    Commented Feb 21, 2020 at 21:45

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