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

Abstract

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.