Distances in Convex Polygons

Summary. One of Paul Erdős’s many continuing interests is distances between points in finite sets. We focus here on conjectures and results on intervertex distances in convex polygons in the Euclidean plane. Two conjectures are highlighted. Let t(x) be the number of different distances from vertex x to the other vertices of a convex polygon C, let $$T(C) = \Sigma t(x)$$ , and take $$T_{n} =\min \{ T(C) : C\mbox{ has $n$ vertices}\}$$ . The first conjecture is $$T_{n} = \left ({ n \atop 2} \right )$$ . The second says that if $$T(C) = \left ({ n \atop 2} \right )$$ for a convex n-gon, then the n-gon is regular if n is odd, and is what we refer to as bi-regular if n is even. The conjectures are confirmed for small n.