The number of distinct distances from a vertex of a convex polygon

Erd\H{o}s conjectured in 1946 that every n-point set P in convex position in the plane contains a point that determines at least floor(n/2) distinct distances to the other points of P. The best known lower bound due to Dumitrescu (2006) is 13n/36 - O(1). In the present note, we slightly improve on this result to (13/36 + eps)n - O(1) for eps ~= 1/23000. Our main ingredient is an improved bound on the maximum number of isosceles triangles determined by P.

💡 Research Summary

The paper addresses a long‑standing conjecture of Erdős from 1946, which asserts that for any set P of n points in convex position in the plane there exists at least one point of P that determines at least ⌊n/2⌋ distinct distances to the remaining points. This is a special case of the general distinct‑distance problem, but the convex‑position restriction introduces additional geometric structure that can be exploited.

Before this work the best known lower bound was due to Dumitrescu (2006), who proved that every convex n‑point set contains a vertex that determines at least 13n/36 − O(1) distinct distances. Dumitrescu’s argument hinges on a counting technique that relates the number of distinct distances from a vertex v to the number of isosceles triangles having v as the apex. Indeed, each pair of points at the same distance from v forms an isosceles triangle (v, u, w), so the total number of such triangles provides an upper bound on the number of repeated distances and consequently a lower bound on the number of distinct distances.

The present note improves Dumitrescu’s bound by a tiny but non‑trivial additive term. The authors show that the maximum possible number of isosceles triangles determined by a convex n‑point set is at most ((13/36+\varepsilon),n,(n-1)/2) where (\varepsilon\approx 1/23000). Substituting this refined estimate into the standard inequality
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