A new proof of Vazsonyi's conjecture

TL;DR

This paper provides a new, self-contained proof of Vazsonyi's conjecture, establishing an upper bound on the number of diameter pairs among points in three-dimensional space without relying on ball polytopes.

Contribution

It introduces a novel proof technique for Vazsonyi's conjecture that simplifies previous approaches and derives a corollary about embedding diameter graphs in the projective plane.

Findings

Maximum of 2n-2 diameter pairs among n points in 3D space

New proof method avoids ball polytopes used in earlier proofs

Diameter graphs can be embedded in the projective plane

Abstract

We present a self-contained proof that the number of diameter pairs among n points in Euclidean 3-space is at most 2n-2. The proof avoids the ball polytopes used in the original proofs by Grunbaum, Heppes and Straszewicz. As a corollary we obtain that any three-dimensional diameter graph can be embedded in the projective plane.