Lionel Pournin

The diameter of associahedra

Abstract It is proven here that the diameter of the d-dimensional associahedron is 2 d − 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.

From Spheres to Spheropolyhedra: Generalized Distinct Element Methodology and Algorithm Analysis

The Distinct Element Method (DEM) is a popular tool to perform granular media simulations. The two key elements this requires are an adequate model for inter-particulate contact forces and an efficient contact detection method. Originally, this method was designed to handle spherical-shaped grains that allow for efficient contact detection and simple yet realistic contact force models. Here we show that both properties carry over to grains of a much more general shape called spheropolyhedra (Minkowski sums of spheres and polyhedra). We also present some computational experience and results with the new model.

Improved bounds on the diameter of lattice polytopes

We show that the largest possible diameter

Constrained paths in the flip-graph of regular triangulations

{\delta(d,k)}

The asymptotic diameter of cyclohedra

δ(d,k) of a d-dimensional polytope whose vertices have integer coordinates ranging between 0 and k is at most

Modular flip-graphs of one-holed surfaces

{kd - \lceil2d/3\rceil-(k-3)}