This post is about a distinct distances problem which I think is quite interesting. As we shall see, solving this problem would also lead to significant progress in characterizing point sets with a small number of distinct distances. Recall that a

*perpendicular bisector*of two points is the line that consists of the points satisfying (where is the distance between and ). For brevity, we will simply refer to these as*bisectors*.We say that a set of points in

*has no repeated bisectors*if for every four distinct points the bisector of and is not identical to the bisector of and (that is, the configuration in the above figure is forbidden).**Conjecture.**

*Let be a set of points in with no repeated bisectors. Then for any the points of determine distinct distances, .*

**Update July 2017:**The conjecture is false, as pointed out by Cosmin Pohoata. A known construction is a counterexample, and I should be ashamed for not observing this on my own. Erdős, Füredi, Pach, and Ruzsa constructed a set of points in with distinct distances for any and no four points on a common circle. Having a repeated bisector requires having four points on a circle, so there are no repeated bisectors. Oh well…For some initial intuition, consider the point sets that determine a small number of distinct distances. I am only aware of two types of configurations that yield distinct distances: the vertices of a regular -gon and lattices.

We can also play a bit with these examples. For example, by taking two regular -gons whose vertices are on two concentric circles, or by taking evenly spaced points on a line (that is, an lattice). All of these examples have a significantly large amount of repeated bisectors.

So far I only managed to find a set with no repeated bisectors and distinct distances. From the other direction, I can only show that every such set determines distinct distances. This is unfortunate, since a stronger bound for this problem would lead to a breakthrough in characterizing planar point sets that determine a small number of distinct distances. Below I describe how this can be done, and also how to obtain simple lower and upper bounds for this problem.

Advertisements