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arXiv · 1207.1266

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

Gabriel Nivasch·János Pach·Rom Pinchasi·Shira Zerbib
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Abstract

Erdős 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.

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Gabriel Nivasch, János Pach, Rom Pinchasi, Shira Zerbib. 2013-03-22. The number of distinct distances from a vertex of a convex polygon. https://arxiv.org/abs/1207.1266

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