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

TL;DR

This paper improves the lower bound on the number of distinct distances from a vertex in convex polygons, advancing understanding of Erd ext{"o}s's conjecture with a refined combinatorial bound.

Contribution

It provides a slightly better lower bound on the number of distinct distances, using an improved bound on isosceles triangles in convex point sets.

Findings

Enhanced lower bound: (13/36 + eps)n - O(1) for the number of distinct distances.

Improved bound on the maximum number of isosceles triangles in convex polygons.

Progress towards Erd ext{"o}s's conjecture on distances in convex point sets.

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.