TL;DR
This paper proves that the diameter of the d-dimensional associahedron is 2d-4 for d > 9, explicitly describing maximally distant vertices and resolving longstanding open problems.
Contribution
It establishes the exact diameter of high-dimensional associahedra and explicitly identifies the vertices achieving this maximum distance.
Findings
Diameter of associahedron is 2d-4 for d > 9
Explicit description of maximally distant vertices
Resolution of open problems from 25 years ago
Abstract
It is proven here that the diameter of the d-dimensional associahedron is 2d-4 when d is greater than 9. Two maximally distant vertices of this polytope are explicitly described as triangulations of a convex polygon, and their distance is obtained using combinatorial arguments. This settles two problems posed about twenty-five years ago by Daniel Sleator, Robert Tarjan, and William Thurston.
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